You can trigger this quest after the completion of Show No Mercy. It turns out that not everyone in the Institute is a native of the secretive base. A doctor by the name of Madison Li (who you met during the Institutionalized quest) used to be a member of the Brotherhood of Steel a decade ago.
Finding her isn't the tough part, thanks to the waypoint indicator. Persuading her to return to the Brotherhood is a whole different matter. If your charisma is high enough to influence her, you can convince her to leave the Institute outright. The more likely outcome is that you'll need to gain her trust by retrieving a holotape from a closed off section of the map.
By following the updated waypoint, access the FEV lab and prepare to take on turrets, Synths, and a very aggressive Assaultron. After a few turns and passages, you'll find a holotape belonging to Doctor Virgil. Return to Doctor Li with the holotape. Then return to Elder Maxson at the Prydwen to complete the quest.