Crimea had at one point been a territory of Genoa, an Italian city state that existed throughout the Medieval and Early Modern eras. It lost control of this territory in the late 1400's.
However, lack of territorial claims didn't exactly halt German march in WWII. Italy contributed significant amounts of troops to Operation Barbarossa (according to Wiki, 235,000 in the start of the campaign). OTL, the Italian army in Russia was destroyed at Stalingrad, but they had been an actually efficient fighting force before then.
What if, after a successful Barbarossa (A vanilla one, like Soviets pushed beyond the Urals, partisans crushed, there are hundreds of Barbarossa Victories on this site), Italy asks for Crimea to be granted to them due to their contributions? Most likely Hitler would laugh the Italians out of the room, but if Italy had crushed Yugoslavia quickly and then not had the fiasco in Greece, Hitler might actually be inclined to at least enter into some kind of negotiation.
What would a Fascist Italian government in Crimea look like? My guess is that since Italy was at least marginally less cruel towards (European) minorities than Germany was, it would not give over the population to Germany. However, I could also see population exchanges where Italians are settled in Crimea, and the local Crimeans are shipped somewhere else (Somalia?) to make the land truly "Italian" so the Germans would have a harder time justifying demanding it from them.
What do you all think?