Institutionalized

Institutionalized is a main story quest and an achievement/trophy in Fallout 4.

Detailed walkthrough
After successfully teleporting into the Institute, the Sole Survivor will be greeted by a voice coming from hidden speakers. It's possible to acquire the holotape needed for Inside Job at this point, as the console is right at the entry/exit point of the Institute. After following the voice through the empty Institute corridors and into an elevator, the Sole Survivor eventually enters a room and finds what looks like their son Shaun in a cell, who is acting scared and confused when the Sole Survivor talks to him. Shaun turns out to be a synth (Shaun (synth)) when a man named Father enters the room and shuts him down with a code phrase. He will then initiate conversation with the Sole Survivor. This conversation reveals that he is your son Shaun and the director of the Institute, as well as several other details regarding the Sole Survivor's situation and the details surrounding his initial kidnapping by the Institute. After speaking with him and agreeing to join the Institute (alternatively stalling by telling him he/she wants to think about it), the player character is free to explore the facility.

Speak to Clayton Holdren, Allie Filmore, Justin Ayo, and Madison Li, who installs a relay into the player character's Pip-Boy, allowing fast travel to and from the Institute.

Bugs

 * Dr. Madison Li may not start conversation with the Sole Survivor. This can be fixed by forcing the quest to the next stage (setstage mq207 200), which will then grant you fast travel access and start the next quest. The same can happen to other staff members. For PS4 and Xbox One, reloading a previous save and trying to initiate dialogue again with the Doctors may help fix this as it did on Xbox One.
 * Father sometimes never shows up. On PC you can use the console (5c337.moveto player) to move Father to your position. The script is waiting for you to talk to the child - so you don't really need to use the console code.
 * You will not be able to speak to the child if you are wearing a power armor helmet with the mod that highlights lifeforms red. Go back to an earlier save, remove that mod or item, and the scenario should run normally.
 * Also Clayton Holdren will not be able to speak to you for some reason. Maybe because you went somewhere else while you were supposed to meet him. He shows up on the pip-boy HUD as being in the Synth Retention Bay, but his location is not reachable.
 * When you finish the goal of "speak with the department heads", and the notification comes up on screen that you may now fast travel to and from the institute, the game freezes and crashes.

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