Editing together parts of previous answers.

The fascist project - as far as what at the time could have been understood as an external racial group (Slavs, Africans, etc.), which is not to say that is was taken as such by default (the definition and identification of races was a controversial issue already, as was the correspondance between biological and cultural terms) - focused mostly in the establishing of a form of political hegemony driven by "cultural superiority". This was a traditional theme developed within the Italian nationalist ideological formulation around the matter of the relations between Italian and Slavs in the Balkans (where the nationalists argumented in favor of the traditional political priority of the Italian urban minorities over the rural majorities) - simply put, the idea that inferior cultures were necessarily and (perhaps) in their own interest subordinated to superior ones. It was then for the inferior culture to adapt to such process of subordination - in a pattern of assimilation that was frequently compared to that of the Roman Empire. As did Mussolini himself in his private speech of October 1938

The racial problem is for me one most important conquest and most important is its introduction into the history of Italy. The ancient Romans were unbelievably racist. The great struggle of the Roman Republic was indeed to know whether the Roman race could aggregate with the other races.

There was therefore no particular accent on the physical extermination of those different groups - since there was, at the root of it, no reason to question the possibility of this "cultural assimilation" process. Those who challenged this view, and suggested a negative answer, were in the Italian context a small minority - and mostly not influential as well as quite disliked.

Notably, the practical means of this assimilation could very well include violent and deliberate actions; consider for instance the systematic killing of livestock (not only the main source of wealth but the only mean of sustenance for large portions of the interior population) during the "pacification" of Libya, which Rodolfo Graziani himself commented in his memoirs observing that “nothing new can be built, unless one destroys […] a past that is no longer keeping up with the present”.

This sort of approach took a turn for the worse during the late 1930s, and more so later on when the smaller fraction which supported the adoption of a more "racial" policy expanded its influence within the state and especially the puppet regime of Salò.

Even so, the idea that Italy introduced a racial legislation due to German pressure is not really something everybody agrees upon - even if it plays well with the common narrative of a Mussolini reduced to Hitler's "lackey", and somehow takes away part of the responsibility of the Fascist Regime. It is a popular view – and you'll find it mentioned in various works on the subject, but there are very few direct or indirect confirmations that the Nazis applied direct or indirect pressure on the Fascist authorities (G. Ciano angrily remarks in his diary soon after the introduction of the racial laws how “The Jews were accusing them of doing as the Germans told. False! The Germans never spoke of that.” - not that we have to believe Ciano, but we also don't have reasons to disbelieve him here, since the point doesn't make him look any better).

What the Germans did – and gets probably conflated with the racial laws of 1938 – is demand, after 1943 and more strongly with 1944, or (to an extent) order the Italian authorities to surrender the Jews for the purpose of their extermination. As a result, of the 7,495 Italian Jews collected by the Nazi occupation forces and the collaborating Salò Regime for deportation in the early months of 1944, 6,885 died within the system of extermination camps.

The context for the development of "racial" laws within Italy goes back to the renewed interest for colonial adventures during the 1930s, with the “pacification” of Libya and the conquest of Ethiopia. That is, the antisemitic laws of 1938 – consistent as they where with the German Nuremberg Laws, and conveniently falling within the time frame of the Italo-German alliance – appear to be more a continuation of an ongoing “segregation-like” policy, originally targeted to the African natives within the Italian colonies.

According to R. De Felice, the choice of switching the Regime's attention towards racial themes came in the context of the general dissatisfaction of the Regime's hierarchy, and of Mussolini especially, with the Italian people, their inadequacy and resistance to a true and complete fascistization. Mussolini, paraphrasing his own complaints, had given them an Empire – and instead of proving worth of it, the Italians were revealing all of their traditional, natural flaws. The formation of a new fascist generation was going to require a strenuous effort (and in this sense, it's very likely that the National Socialist example may have influenced the many smaller and larger leaders who chose “to re-establish their fascist purity” - quoting G. Bottai by memory – by going on a "pilgrimage" to Germany around 1938) since the current Italians were certainly not up to the task (and by the mid 1930s Mussolini was already certain of the inevitability of a future European conflict, by 1942-43 at most).

Soon after the conquest of Abissinia, a prominent Fascist Gerarca and leader of the “intransigent” current, Roberto Farinacci, had traveled to the new colony and wrote back to Mussolini (April 24^th 1938):

Our [con]nationals are not well guarded and give the natives a far less than imperial display. Many [unfortunate] episodes are caused by their behavior, either gross, indecent or offensive [...] absolute lack of understanding and respect for the traditions of the natives, dreadful attitude towards the native women, loan demands towards well off natives, indecent ways of living and dressing in comparison to the natives.

It's worth pointing out that what Farinacci intends for "respect" is a good measure of distance from the "natives". Or paraphrasing: how could the Italians claim mastery of other peoples if they couldn't even stand tall above... well... As Mussolini himself explained in a private speech of October 25^th 1938:

[…] when a people takes conscience of its race, it does so in contrast to any other race – not one alone. But we did so only in opposition to the “Camites”, which is to say the Africans. The lack of racial conscience has had severe consequences in the Amara [a region of Ethiopia – likely a synecdoche]. […] when they saw that Italians went more raggedly than themselves, that they lived in “tuculs”, that they took away their women; [the natives] said to themselves: this is not a race that brings us civilization! […] To preserve the Empire it is necessary that the natives have absolute, undisputed understanding of our superiority.

This core concept had found its way in Mussolini's mind somewhere during the previous years – and we know that with some certainty since, according to his habits, Mussolini had begun to favor the publication of racist and antisemitic works since around 1936 (that is before the Italian relations with Germany grew close enough to warrant a direct ingerence – as I mentioned, there were intellectual contacts and growing ideological contiguity, for instance with “experts” exchanging their views on the matter on specialized press, but there is no evidence of political or diplomatic pressure during 1938-39), for example Paolo Orano's “The Jews in Italy” published in April 1937. There is little doubt that the antisemitic turn around 1938 had a large portion of instrumentality in Mussolini's mind – but that did not mean it was unplanned, nor entirely unwelcome. Rather, the view that ideological positions had to serve as instruments was one of the traits that more strongly separated Mussolini from Hitler. As Mussolini had attempted to explain in a letter to the newly appointed German Chancellor in March 1933:

Every regime has not only the right but the duty to remove from places of authority any element not entirely trustworthy, but to this end it is not necessary, rather it could be damaging, to bring on the racial ground – Semitism and Arianism – what is instead simply a measure to protect and develop the revolution.