Why doesn't the U.S. Air Force oversee all aviation for the U.S. Armed Forces?

Every branch of the U.S. Armed Forces (Army, Navy, Marine Corps, Air Force, and Coast Guard) has aviation, but since we have a branch that specializes in aerial warfare, the U.S. Air Force, why don't they oversee all aviation? For example, the Navy oversees the waters outside U.S. territory, the Army oversees land based operations, the Marine Corps oversees all amphibious warfare, and the Coast Guard oversees the waters within U.S. territory. If this is so, why doesn't the Air Force oversee all aviation for the U.S. Armed Forces?