But if Omega is Mafia, why didn't he pretend to be a role without citizenship? Why did he start shouting that he was a citizen of the USA, if the opposite is obvious? I don't understand.
He got nervous and gave himself away. There was clearly no team nearby to discuss things with, and he needed to defend Hermit, otherwise, when votes were cast against him, Omega's ridiculous excuses wouldn't have looked good. This was his only chance to save his teammate, and he saved him. He sacrificed himself :)
