Institutionalized

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

Quick walkthrough

 * 1) Enter the elevator
 * 2) Meet the division leaders

Detailed walkthrough
The teleportation is a success. Listen to the voice directing the Sole Survivor. Oblige the voice, and one soon finds a man named Father, who runs the Institute. 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. Access Institute Bioscience and meet Doctor Clayton Holdren.

Find Chief Engineer Allie Filmore in the main concourse. Access Institute SRB and meet acting director of the Synth Retention Bureau, Justin Ayo. He doesn't seem to trust the Sole Survivor. Access Institute Advanced Systems and meet Doctor Madison Li, who installs a relay into the player character's Pip-Boy, allowing fast travel to and from this location.

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.