Irish Gangster
The Gangster knows the Boss and the common aim of the team, but gets no night checks. Their strong side is working with public information and keeping the team alive.
The first day
Build your own argument. Blindly repeating the Boss's position creates a visible link. Also avoid an artificial permanent conflict: it can end in a vote against your own.
Hunting the Italians
Truly exposing a rival is good for the Irish, but may make you "too useful to the city" and raise the suspicion that you belong to the other gang. Do not count on exposing an Italian automatically earning trust.
If the Boss is alive
Use day observations to help the team judge targets, but do not claim to know the results of the Boss's checks if the game procedures do not let them be passed to a partner. The way of coordination must fit the night rulebook.
Do not repeat every speech of the Boss. Create an independent line that can be checked, look for the Italians and make sure that paired defence does not give the team away.
If the Boss has left
Move to autonomous play: keep your cover, weigh the numbers of the city and the rivals, and choose the night target. Shooting goes on under the approved rule.
Keep playing and shooting, but the search for the Sheriff stops. Reconsider old suspicions and act on public facts.
Typical mistakes
Playing only as the Boss's bodyguard; voting against everyone; presenting versions as knowledge; thinking that a lone Gangster has already lost.
Do not count the Italians as allies by default. Do not pretend to know the other gang when you do not.