TO ANYONE COMMENTING WITH EXTRA QUESTIONS: I keep getting alerts about it but for whatever reason I'm not being allowed to read the questions. As soon as that's worked out I'll do my best to answer them

Disclaimer:(Likely to need editting later as this was written on mobile during my lunch break, but I wrote a reasonably extensive paper on this exact topic a few years back and still have access to all the sources)

Essentially the role played by the Mafias during the invasion was non existent as by that time Mussolini had mostly gotten them under control and put into prison.

That lasted until about 5 minutes after 'liberation'. (I really hate the term liberation as it's extremely non-objective and propagandistic so I will not be using it)

Due to the ideological nature of the conflict, particularly with fascism renowned for its imprisoning of political rivals, one of the first steps taken by the incoming forces was to open up all the prisons and release local political figures and notables. Many times these released individuals were the big bosses of local or major crime syndicates, and they rapidly returned to their former positions as the chaos of having multiple armies spreading throughout the countryside without any interest in governance gave them a ton of opportunity for black market profits.

These figures were largely neutral in the war itself though many got scooped up as impromptu administrators due to the insular nature of many south Italian communities. After the war in fact there was a immediate rush of former Mafia members and leaders in local political offices governing all throughout the south.

During the late war when the priority shifted from 'winning' to 'rebuilding' many mafia leaders were also grabbed to help(or just took the position by force) distribute relief supplies. Needless to say a significant portion of them went towards mafia revenue.

This position as redistributors of the local wealth and relief force workers made them extremely popular, and the inward looking communities of the region were largely supportive of the efforts of these people.

They would largely become nominal supporters of the Christian Democrats under DeGasperi due to their shared opposition to communists, and would promote this support to the extent that the south would become a bulwark against the more left leaning north of the country, deepening political divides (which were already enormous in this young country) in a way we still feel today.

Overall the Mafia played a moderate to significant role in administration of Southern Italy following the arrival of the Allied forces, one which they leveraged to acquire greater wealth and local power. Both of these they would then use to acquire legitimacy as the desperate need of the Italian Government to acquire able administrators led them to ignore mafia credentials.