Note: Note: This quest will exclude you from all future missions with The Railroad.
This quest might be tactical in concept, but its execution is direct and forceful. Speak to Lanser-Captain Kells after Blind Betrayal. He'll have orders to kill key members of The Railroad. By accepting this quest, you'll have made yourself an enemy of The Railroad, so any privileges you might have with them, like the ability to fast travel to their HQ, will be revoked. It's not much of a problem, since you can just fast travel to their base's perimeter at Old North Church.
You should by now know where their hideout is within the church's catacombs, but if you've managed to avoid The Railroad up to this point, you can refer to Road to Freedom for specifics. Note that Doctor Carrington will be part of the welcoming committee. He's also one of your primary targets. You won't have to solve the dial puzzle this time around as The Brotherhood will have already planted an explosive for you. You simply have to activate it.
Once inside the headquarters, prepare for a close quarters fight within the base's cramped spaces (using VATS helps). Your kill list includes Glory, Tinker Tom, Deacon, and Desdemona, but you'd might as well kill everyone in HQ, based on who's easiest to target at any given moment.
The only loose end left is the Railroad's robot, P.A.M.. The Brotherhood hopes you can reprogram P.A.M. to work for them. You can do this by accessing the mainframe (in a room adjacent to the main area of the headquarters). Otherwise, you can just destroy her with your weapons, much to the mild disappointment of The Brotherhood.
Report back to Kells to complete the quest.