If you have a player that is voted into exile, must that player immediately reveal to the rest of the group whether they are actually the betrayer?
When the players vote in favor of exiling a player, the exiled player must immediately draw 1 exiled secret objective card. This text of that card follows the following template:
If you are not the betrayer: Reveal your secret objective to all players and remove it from the game. Your new secret objective is now the following: [...]
If you are the betrayer: The 1st goal on your secret objective has changed to the following: [...]
So you must implicitly reveal that you are the betrayer by keeping your original secret objective card.