It is hard to answer a question about extent without a given factor (money, manpower, influence). Let me try to answer generally because the mafia's relationship with fascism changed over the course of its short existence. We know the mafia outlived fascism see the murder of Falcone, the Clean Hands Affair, the kidnapping of Aldo Moro, the French Connection, etc...

In the beginning, the mafia infiltrated Fascism like it always had infiltrated institutions of power. Upon one visit to Palermo, Mussolini had a mob boss as essentially a bodyguard as his security detail thought this would keep him safe. Mussolini did not take kindly to this as you can imagine a fascist did not like the idea that he did not have complete control over his populace. Sicily is acknowledged to be by far the least sympathetic region to fascism in general. The why is debated. After this event and some others, there were calls by fascist leaders to crack down on the mafia. It is hard to be a law and order party with rampant organized crime operating under your nose. Sicily also suffered from widespread banditry at the time. One such group led by Salvatore Giuliano who is suspected of having mafia ties became quite the folk hero.

Anyway, Mussolini sent a force down into Sicily to root out these mafia dens. One of his most famous captains Cesare Mori essentially completed large dragnet operations which arrested Sicilians some of whom were undoubtedly mafiosi and others who were just Sicilians. This harsh crackdown essentially put the mafia into hiding and reduced its numbers to some extent. These are the dark days for the mafia.

The Allies would essentially breath new life into the mafia upon retaking the south. They needed a powerful group of people to connect them to resources and there was really no one else around. Some historians (Pantaleone) place the revival of the mafia squarely at the hands of the Allies.

Reece, Jack E. “Fascism, the Mafia, and the Emergence of Sicilian Separatism (1919-43).” The Journal of Modern History, vol. 45, no. 2, 1973, pp. 261–276. JSTOR, JSTOR, www.jstor.org/stable/1875676.

The nature of the relations of the Fascist Regime with the Sicilian Mafia can only be understood by taking into consideration the degree of contiguity between the "Mafia authority" and the "State authority" in the Island during early XX Century. Those two aspects of public life did not exist in true opposition to each other - rather, without even being alternative, they translated fluidly into each other.

It was common at the time to blame the Mafia phenomenon on the "resistance of certain strata of the population, who reject the deep reasons of the unity and centrality of the state" - as explained by Catholic intellectual Pietro Mignosi. It was, that of the resistance against the ideas of national unity, almost a trope of the post unitary approach to the Southern problems - not that it should be entirely disputed, but one should handle the idea with much more care than it was done in the 1920s.

Regardless of the accuracy of the evaluation, this problem was almost obviously a central one for the Fascist Regime, which claimed to hold the keys to restore the true authority of the State, unitary and central, that had been compromised by the weak policies of the liberal state. And yet, the authority of the state has to move through the concrete institutions of the state: local administrations, city councils, praefects, police authorities, law courts, etc. Where the presence of the Mafia grows too strong, then the authority of the state reduces de facto to the authority of the Mafia.

Hence what already at the end of XIX had been highlighted as the key element of the "mafioso mindset" - the idea that "Mafia was necessary if one wanted to amount to anything in life". Again with Mignosi: "The mafioso gains city councils, provinces, parliament seats, not because he has faith in the state [...] but because he feels the need to neutralize [the state's] authority; he needs to feel as free as possible in the realization of his true aspirations. In this sense the mafioso is truly a-political: he has no understanding of parties or ideologies. And yet at the same time, he feels more political, absolutely political, because [he is] above politics."

Here the material authority of the state does not rest on the ideal authority of the state, but in the material means that allowed to disregard or outright refute the ideal authority of the state. And thus to replace it. Wrote Cesare Mori:

When a mafioso manages, by means that one needs not discussing, [...] to put together some, or plenty of god given wealth [...] then he softens up, he grows fond of all humankind and he... seeks redemption. He does that all by himself, that is he absolves himself [...] and crosses with all the apparatus (lands, houses and livestock included) under the protection of the state's law [...] He becomes a man of order, rigidly, ferociously - dare say - a man of order.

It's not a surprise that fascism enjoyed a certain degree of sympathy among the Sicilian Mafia. It's not a surprise that fascism created concern and insecurity among the Mafia either. The phrasing is not accidental - it's not Fascism which reacted to the presence of the Mafia, rather the Mafia reacted to the rise of fascism.

In the late XIX Century, early XX Century, the Sicilian Mafia was very much tied to ownership and control of the land - the great landowners had reworked the local authorities into appendages of their own. Fascism found its affirmation as a mass movement first and foremost in the context of the anti socialist reaction of the agrarian world during 1919-21.

This reaction came through the appeal that the new themes of self-affirmation, individualism, honor, violence, a modicum of eversion of the law especially when aimed at the restoration of a "true order" alternative to that of the liberal state; a fascination that all those themes exerted on a mixture of veterans, partially employed land workers, younger generations discontent with the perceived overbearing presence of the centers of power (whether the socialist leagues or the land owners themselves). Those themes of "fascist appeal" are not that far removed from those of the "Mafia appeal".

Fascism did more than attack the existing Mafia structure - for the first time, in a way, it challenged the assumption that there was only one pathway to amount to something. One could very much join the action squads instead - even, why not being both? If you could be mafioso and mayor, mafioso and senator, you certainly could be mafioso and fascist.

In the immediate aftermath of the Great War, the old generation of Mafia began their decline - "the time was serious [...] there was no discipine, the young ones were no longer willing to listen" according to Mori's summary. On a very practical note, the action squads offered the exact same blend of protection and threat that was the traditional staple of the old agrarian mafia - and yet, they also gave rise to the hope that, in the long run, those unruly "fascist mafiosi" could be assimilated into the state, to accept its unity, becoming part of that authority and no longer an exception to it.

On this aspect, Fascism failed. Not generally for lack of trying - Mori's iron fist reputation is legitimate and the Mafia had suffered a crisis in its more apparent organizations already in the early 1920s (as testified by the contemporary sources and by Mori himself). But as fascism grew strong under the patronage of the agrarians, the political contiguity between Mafia and State persisted. Mori himself observed that it was "unfair to attribute the definition of mafiosi to the great land owners of the Island" - those, that is, who had accepted to support the new regime outside of the more clumsy mafia-style proclamation.

Unfortunately, as we discussed, the boundary between Mafia and State maintained its fluidity because, simply put, Fascism had to grow into its "national clothes" and those clothes included the Sicilian Mafia. So that a large and effective action was carried off to suppress the new generations who had not embraced the Fascist Regime (and even a few of those who did - one more notable perhaps Alfredo Cucco), but left somewhat untouched the higher, more stable and more organic expressions of authority in the Island - the Alta Mafia.

Meanwhile, as the structural causes of the Mafia phenomenon remained largely untouched, the ground remained set for the rise of post '43 Mafia and the return of those of the younger generations who had escaped or weathered Mori's storm.

Incidentally, within Sicily this process largely coexisted with the more general tendency to develop a sort of internal power struggle between the elements of the "fascism movement" - that is, in their basic form the actions squads and their local expressions - and the direction organisms of the Regime. Which makes the whole thing more layered and complex than my short summary may suggest.

More so, for this I made largely use of a somewhat outdated source: S.F. Romano's Storia della Mafia [1966]. It gives an interesting perspective on the issue, and I thought worth putting this answer togerher, but is shows here and there its age.

Follow up question: after liberation of southern Italy (Naples, Calabria, Sicily, the areas in which the mafia had the strongest presence), how did they deal with the allies and how did they interact with the Fascist Italian Republic in the north? Did they pick a side? Did they actively support the Fascist government or one of the resistance movements?

Not to discourage further discussion, but this topic has been addressed quite a few times.