A factional ability can be used by all players in a faction. This is in contrast to an individual ability which can only be used by a certain player.
The Mafia factional kill ability can be potentially used by all mafia players so even if only one mafia player remains alive he can still use the factional kill to eliminate players. Generally, the game host would ask all mafia players to reach a consensus on a kill target and which mafia player will perform the kill. However, sometimes the game host might assign the factional kill to one mafia player (Usually the Godfather) and give it to other mafia players as the mafioso start getting eliminated.
If the Town has a factional ability, because of the uninformed nature of the Town, the user of the ability is generally first come, first serve. If the Town has a one-shot vengeful kill factional ability then the first town player lynched will be the one who will use the factional vengeful kill.
In the case of solo factions like the Serial and solo independents, the ability by default becomes individual because the faction has only one player.
In the case of a Cult, the recruit ability is generally individual and tied to the Cult Leader because it usually ends up being overpowered if it is factional. If the recruit is factional, then the game host lets the Cultists reach a consensus on which player should be recruited and which Cultist will perform the recruit. In such a case scenario, the death of the Cult Leader is not a big blow to the Cult as another Cultist will perform the recruit in his stead.
What result does a Cop get if he investigates an Indie?
If a Cop investigates an Indie, The Cop should not get a Guilty result as Guilty is mainly used for anti-town players and the Indie is not an anti-town player.
If a Cop investigates an Indie, The Cop should not get an Innocent result either as Innocent is mainly used for pro-town players and the Indie is not a pro-town player.
The Cop should get a "Neutral" result when investigating an Indie. This is because an Indie is not involved in the victory or loss of Town and as such is a neutral player.
Giving Guilty results on Indies to Cops is an outdated method and can affect game balance. An Indie can both help and hinder the Town depending upon his choice. Thus "Neutral" is the most appropriate result and should be used by the game host.
More to come.