Wikipedia indicates this is the case:
However, since the submarine tender Emory S. Land, based in La Maddelena in Sardinia, changed homeports to Bremerton, WA, the fleet has just one permanently assigned ship, the Mount Whitney. The fleet typically has a number of frigates and destroyers assigned, as well as those vessels transiting between the East Coast and the Suez. Additionally, since 2005, Sixth Fleet ships have increasingly been operating around Africa, particularly in the Gulf of Guinea.
Our fleet is shrinking. So the former battlefield of a potential World War III against the Soviets took a major hit. But the reputation lingers, making even many Americans think our fleet sails supreme there.
This would have to be run from Italian air bases. Apart from the question of the wisdom of it.