A Saulus is originally an anti-town role, but when it should die (either by elimination or other kill), it stays alive and converts to the Town's side instead. The Town is told nothing except that the elimination failed.
Saulus is more difficult to implement than Judas because simply taking the above as the only things different about Saulus, the Saulus role can simply tell everyone their ex-scumpartners as soon as it converts. The only really feasible way to do this is to not tell the Saulus role who their partners are, essentially making them a Mafia Traitor prior to their first death.
The other way to implement Saulus is tell them who their scumpartners are, but forcibly replace them upon death. If the Saulus role is eliminated, the forced replacement is a dead giveaway that their alignment changed. If the Saulus role is killed overNight... it may still be a giveaway, but more to the scum than anyone else. The advantage of this approach is that Saulus can be told who their partners are but still perform in good faith as a Townie later.
A Saulus who converts on elimination should lose if the main Mafia group is dead, to avoid giving them a potentially automatic win in such a scenario.
With the above implementation in mind, it is not a far step to say that the Saulus aspect can be Hidden until they convert. The Saulus role receives a Mafia Traitor Role PM in this case, using the Traitor variant that does not know who the scum are.
Saulus may also appear in games with Judas, muddying what kind of role caused the elimination to stop.
In Greater Idea Mafia and variants, Saulus only turns to town on elimination, not if it is nightkilled or dayvigged.
Use and Power
Without any knowledge of who the scum are, Saulus is more likely to win by acting like a one-shot Deathproof Townie - it is more likely to die somehow before the scum attain a majority than to win before conversion. With that in mind, that's basically what it amounts to.
With knowledge of who the scum are, the Saulus role's optimal play becomes much hazier, as it may still be better for them to play like a Townie from the beginning - but this time the Saulus knows who the scum are.