Like most Institute quests, this one begins by talking to Father. Go talk to him and he'll ask how the meeting went. Answer however you wish. Now, if you have contacted the Railroad you will get some text here calling your allegiance into question a bit, and you can even outright tell Father they are your allies but no matter what you do your goal (from Father, who is outright commanding you at this point) is to "Kill Desdemona".
Warning: This is the deciding point between the Railroad and the Institute.
Make your way to the Railroad and kill everybody you find dwelling down here. Be sure to bring healing items and a companion other than Deacon for this fight. There is no need to actually talk to Desdemona, but you can tell her you are here to kill her nonetheless. It would be better though to get hidden and get in sneak attacks to thin the herd. After you kill Desdemona, you will be told to "Eliminate Railroad Leaders", which include Tinker Tom, Glory, Doctor Carrigan and Deacon. Note that if you have Deacon as a companion out waiting for you at a Settlement, you will have to go there and kill him separately as well. There is really no notable loot to be had, but Desdemona does have the "Railroad HQ Key" and a Railway Rifle you should grab. Warp back to The Institute once you are done.
Talk to Father to report your success and he will sympathize a bit and tell you that the Brotherhood is next. Because OF COURSE they are. Well, at least you are already enemies with them. He will go over how the Brotherhood is quite a bit bigger than just the Railroad was, with other factions out in the Wastes. As he talks, this quest will officially end and the next quest named "Airship Down" will begin. Father will ask you to go see Doctor Watson to go over his plan on how to deal with the Brotherhood, but we'll cover that under "Airship Down" .
|For exterminating the Railroad leadership||590 XP|