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 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 Virgil's disappearance from his laboratory.

In order to ensure Doctor Li's trust, you agree to recover evidence regarding an accident that supposedly killed her colleague, Doctor Virgil. His former lab is in a sealed section of the BioSciences Division. To access it you can: (Master) unlock the FEV lab entry terminal by the laser grid gate or (Novice) unlock the FEV lab door.

Venture through the FEV lab, which has automated turret security, synths and an assaultron. Locate the holotape (Brian Virgil personal log 0176) at the laboratory.

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.