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 to convince Madison Li to return to work for the Brotherhood of Steel.

Detailed walkthrough
During your infiltration of the Institute, Elder Maxson orders you to make contact with a Madison Li and convince her to work on a "special project" for the Brotherhood of Steel. Find her within the Institute in the Advanced Systems division and let her know you've been sent to find her (or you can be direct and tell her the Brotherhood sent you).

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

In order to ensure Doctor Li's trust, you 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 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 several turrets and and a level-dependent assaultron which may be hard to fight for low-leveled players 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, you can also pick up Virgil's serum, on a nearby table, that must be returned to him as part of a side quest.

Return to Doctor Li and hand over the holotape. Then continue with the main quest Institutionalized. Once you leave 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 courser 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.