From Within

From Within is a Brotherhood of Steel main quest in Fallout 4. The quest takes place in the Institute where the player character is tasked with convincing Madison Li to return to work for the Brotherhood of Steel.

Detailed walkthrough
During the infiltration of the Institute, return to Elder Maxson and speak with him to start the quest. He orders the Sole Survivor to make contact with Madison Li and convince her to work on a "special project" for the Brotherhood of Steel. Return to the Institute, find Doctor Li within the Advanced Systems division and let her know that the player character has been sent to find her (or tell her directly that they are sent by the Brotherhood.)

There is also a speech check (Medium / Hard) where she can be asked if she is happy, and complete a series of difficult, fast-talking challenges to convince her. If she agrees, one can skip to the final objectives of the quest. Otherwise, the player character can agree to find evidence about Brian Virgil's disappearance from his laboratory.

In order to ensure Doctor Li's trust, one must agree to recover evidence regarding an accident that supposedly killed her colleague, Doctor Virgil. His former lab is in a sealed section of the Bioscience Division. There's a back entrance with a master lock terminal in the main room of the Bioscience lab leading directly to the main room (avoiding all of the security), and also the main entrance to it found at the end of some seemingly abandoned storage halls and rooms, behind a novice-locked door. Once in the FEV lab, travel through it to the end room, where the Brian Virgil personal log 0176 holotape which must be returned to Li is located. The lab will be defended by traps, several turrets and a level-dependent assaultron which may be hard to fight for low-leveled player characters due to the sealed environment but is otherwise uninhabited. Once in the last room, locate the holotape, which can be found by an active terminal. While there, one can also pick up Virgil's serum, on a nearby table, that must be returned to him as part of Virgil's Cure.

Return to Doctor Li and hand over the holotape. Upon next leaving the Institute, return to the Prydwen and inform Elder Maxson.

Bugs
When this quest is active at the same time as Institutionalized, you may not be able to receive the Pip-Boy fast travel upgrade from Doctor Li, which prevents you from leaving the Institute; attacking/killing any scientist seems to circumvent this, as it causes you to be kicked out of the Institute.