I searched the forum for variations along the lines of the following scenario, but found nothing:
What if Marshal Badoglio and Victor Emmanuel III are killed on 8-9 September 1943 while attempting to flee Rome. Three days later, because of intervening butterflies, the Gran Sasso Raid fails, with Skorzeny killed. Mussolini is evacuated under heavy guard to Brindisi, then Algiers and thence to Britain, where he is imprisoned in the Tower of London. He reportedly befriended Rudolf Hess.
Now, in such a scenario, I understand that the RSI likely wouldn't be formed, as Hitler didn't think highly of the other Italian fascists. So Italy will be under put under a German military government. There will also be much, much more Catholic/Conservative resistance than IOTL, since the Germans killed "their" king. (In fact, it's unlikely that the monarchy would be abolished after the war ITTL.) The Allies will also need to organise their own military government, something for which there was no need IOTL. Finally, I wonder what the military impact of this PoD could be, especially whether it can lead to more territory under Western control at the end of the war.