The post-war situation in Italy was extremely complicated but in short, the Allies mostly allowed Italy to sort itself out. With one major exception. Reason being is that the Allies signed a peace treaty with an entity which had not been defeated.

The root cause of the situation c as n largely be found in the Armistizio di Cassibile of 8th September 1943, the armistice signed by the new government of Marshall Badoglio, at the behest of the King, and the Allies. As a result of the armistice, the legitimate government of Italy became a co-belligerent and was to continue the war at the side of the Allies. This in itself would have meant that it would have been difficult for the Allies to impose similar terms on Italy as it would eventually be the case with Germany and Japan.

To further complicate matters, power shifted very quickly from the King and Marshall Badoglio, to the Comitati di Liberazione Nazionale (CLN) , the alliance of parties fighting against the Germans and Mussolini's Repubblica Sociale. The CLN was quick in seizing political power in both Allied and German occupied Italy. In the south this meant that from April 1944, the second Badoglio government was a CLN government and from then on to wars end the three subsequent PMs were all from member parties of the CLN.

In the North, this meant that the only non-fascist Italian political presence was the CLN. The parties which formed the CLN all had their own partisan 'brigades' and took an extremely active role in fighting both Germans and fascists. And the complication here for the Allies is that this was extremely beneficial to them as it tied up a not indifferent number of Axis troops and because of this, the Allies were in turn providing vital supplies. The importance of the CLN in German-occupied Italy was such that after the general strikes and insurrection of late April 1945 which effectively ended the war in Italy, German commanders often had to surrender to the CLN rather than the Allies (which often woul arrive with one to two days delay).

At this point, it's worth explaining which parties formed the CLN as it will matter for what happened next. The CLN was formed by the Communist Party, Socialist Party, Labour Democratic Party, Liberal Party, Action Party and Christian Democrats. These parties were split in two blocs, a compact and tightly knit left-wing bloc (Communists, Sociakists and Action Party) and a looser less cohesive right-wing bloc (Liberals, Christian Democrats and Labour).

The importance of the political make-up of the CLN is important as it dictated it's attitude to the King and the monarchy as a whole. The right-wing bloc favoured a more measured approach to the monarchy and pushed for creating a situation were the monarchy would remain a factor in a Italian politics with a discussion on its future after the war. The left-wing bloc, instead argued for cutting out the King and the monarchy from political process straight away. In any case, the CLN initially formed United front in asking for the King to abdicate (which he eventually did in June 1944); things changed a little with the svolta di Salerno when the Italian Communist Party made the decision to put on hold its political project in order to participate fully in a government of national unity.

At war's end, the issue of republic vs monarchy came up again and a referendum was held on 2nd June 1946. Vote for a republic narrowly passed with 54.3% of the vote on a turn out of nearly 90%. This paved the way for the beginning of the Republic with a new constitution coming in force in 1947.

To go back to your question, this complicated things for the Allies beyond what the armistizio di Cassibile had done. The Allies were no longer deking with the King, who had appointed Mussolini and could be somewhat be held culpavle; they were now dealing with a republican government formed by parties which had taken an active role in fighting the Axis, which received supplies from the Allies and which had been banned under Mussolini's regime. The result was that although Italy was to pay reparations, reduce the size of its military and abandon its colonies, the Allies did not impose an occupation force or administration for a set period of time. Agreements were made for the US to have military bases but that is the extent of Allied presence in Italy after the war.

There was, however, an exception: Trieste. This was a major point of contention as the Yugoslav government of Tito claimed Trieste as part of its wider claim for Dalmatia. On the other hand, the Italian government considers Trieste integral part of Italy. The result was that the Free Territory of Trieste was formed and admistered by an Allied occupation force until matters between Italy and Yugoslavia could be solved. Eventually, the Allied administration was dissolved in 1954 and the majority of the Free Territory returned to Italy.