Other choices being Cubans, Russians, Jealous Husbands?
After Kennedy was shot but announced that he was still alive:
“He (Fidel Castro) came back, sat down, and repeated three times the words: “Es una mala noticia.” (”This is bad news.”) He remained silent for a moment, awaiting another call with further news. He remarked while we waited that there was an alarmingly sizable lunatic fringe in American society and that this deed could equally well have been the work of a madman or of a terrorist. Perhaps a Vietnamese? Or a member of the Ku Klux Klan? The second call came through: It was hoped they would be able to announce that the United States President was still alive, that there was hope of saving him. Fidel Castro’s immediate reaction was: “If they can, he is already re-elected.” He pronounced these words with satisfaction.”
Don't forget my favorite -- the French.
I don't believe there was any conspiracy, but Oswald crossed paths with so many Soviet, Cuban, and US operatives of one agency or another that conspiracy theories are inevitable.