I think that Godfather should be removed from the investigation pair "Ml, Mayor...etc".
The reason is that the setups that grants detection immunity easily stomp the ones that does not, making it almost the official setting. There is no way for new players to tell how Investigator works under detection immunity, unless being told by someone who already knows (who probably made the same mistake).
As for Godfather new investigation pair, I think it should be the "Cit, Surv...etc" regardless of immunity status. If you somehow feel pitful for the minority of the setups that excludes immunity, Sheriff simply can detect Godfather in these.