Italy was the only great power that failed to develop carrier based aviation during the interwar period, although it seems it could have developed strong capabilities if it had decided to go that direction. Starting in late 1919 Italy considered completing one of its Francesco Caracciolo class battleships as an aircraft carrier, but a tight budget situation resulted in Italy deciding not to complete any of the ships in any form (France completed Béarn, a Normandie class battleship, as an aircraft carrier). Italy did have extensive experience with military and civilian seaplane based aviation (the Macchi M.C.72 remains the world's fastest piston powered seaplane) and that could help with development of its carrier aviation. Building an aircraft carrier could also change the doctrine of the Italian Navy, as there was considerable debate over which direction it should go after World War I. Some wanted battleships, some wanted aircraft carriers and submarines, and some wanted a mixture of the two (source). While Chief of the Naval Staff Admiral Paolo Thaon di Revel argued that "the development and use of aeroplanes in wars on our seas and along our coasts is today the most essential element of national defense," Minister of the Navy Admiral Giovanni Sechi felt that aircraft carriers weren't necessary in the enclosed waters of the Mediterranean and that land based naval aviation would be sufficient (source). Sechi scrapped many older Italian ships and canceled others, including the Francesco Caracciolo class, to help free up funds for new construction, so there may have been some funding available for an aircraft carrier conversion within the historical budget. How might the Italian Navy have looked by the 1930s if the pro-aircraft carrier faction succeeded in having Francesco Caracciolo completed as an aircraft carrier conversion? Might it see service supporting military operations in Libya and Ethiopia?
It's only possible to project air power within a few hundred miles of an air base or aircraft carrier. Italy can cover extensive areas with land based aviation assets, but assuming a 500 mile combat radius areas such as Gibraltar, Spain, Southern France, Greece, Egypt, and Crete would be difficult if not impossible to cover. Aircraft carriers are mobile air bases too, so they they could help reinforce Italy's overseas holdings more easily than land based aircraft could be flown or shipped to them. Here's a map that illustrates this (source):
Oh course, but that doesn't mean Italian carrier based aviation would be irrelevant. Its Littorio class battleships were some of the most powerful warships in the world and proved a challenge for Royal Navy operations in the Mediterranean during World War II. It could be the same with Italian aircraft carriers, perhaps with it building a pair each of an advanced battleship and aircraft carrier design. Italy also had extensive military dealings with foreign countries during the Interwar Period, and knowledge and technology from its own experience would likely influence other navies as well, including the Soviet Union. If Italy ends up in World War II and has some of its aircraft carriers taken as reparations they could even influence Soviet naval doctrine.