Institutionalized

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

Detailed walkthrough
If the teleportation is a success. Listen to the voice directing the Sole Survivor. Oblige the voice, and he soon finds what looks like his son Shaun but turns out to be a synth "Shaun (synth)" and a man named Father, who runs the Institute, the conversation with him reveals that he is your son Shaun as well as several other details regarding your situation. After speaking with him and agreeing to join the Institute (an offer can rescind at a later date), 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.
 * 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.
 * Also Clayton Holden will not be able to speak to you for some reason. Maybe because you went somewhere else while you were supposed to meet him.