This lengthy, pivotal Railroad request becomes available after completing The Molecular Level. Having learned about your infiltration into The Institute, Desdemona will ask you for a favor involving a meetup with an Institute insider named Patriot. After talking to Desdemona, speak with Tinker Tom, who will give you a holotape and the first steps of the assignment.
Once inside The Institute, access any terminal and insert the holotape. Look through the tape's contents to find the encrypted message that will lead you to Patriot's whereabouts. Meet him in the Advanced Systems maintenance room. Patriot's real name is Liam Binet and he will lead you to Z1-14, a Synth. After this conversation, talk to Liam again, who will express the need for a specific, very old password. Then return to Railroad HQ.
Speak to Desdemona to give her a status update. She will direct you to P.A.M.'s computer. Doing this will give P.A.M. enough information to deduce the likely location of the password. This will lead you to Cambridge Polymer Labs. Once there, go from the lobby to the lab, using the clean room in between.
Once in the lab, follow the waypoint past the locked door and to the indicated terminal. Retrieving the password will cause a security lockdown. The act of removing this lockdown is actually a side quest called Cambridge Polymer Labs. The quickest way to solve this is to remain at the terminal and select the Facility Defense Systems override option. You can now leave the lab by backtracking to its entrance.
Travel back to Railroad HQ after exiting the facility and speak with Desdemona. Now teleport to The Institute and meet with Liam to give him the password. Then meet with Z1-14 for an update. Z1-14 will need 24 hours to ponder the next move, so either distract yourself for a day or advance time by sitting or sleeping.
Z1-14's game plan is to assault a group of guards at a nearby tunnel and storage room. Other Synths will participate in this surprise attack, and you have the option to support them by transferring some of your weapons into a container indicated by a waypoint. When you're ready, go to the storage room and assist the Synths in killing the guards.
Have another conversation with Z1-14, who will then advise you to keep a low profile by spending time with Father. To complete this task, you'll need to complete two Institute quests, Mankind Redefined and Mass Fusion.
After completing those quests, a discrete Synth messenger will urge you to return to your room. You'll find Z1-14 waiting there for you. He happens to have gotten wind of a Brotherhood assault on Railroad HQ. Go there to update Desdemona, and be ready to defend yourself and The Railroad. This will trigger the next faction quest, Precipice of War.