From Within Brotherhood of Steel Quest
This quest will be triggered or can be aquired when you complete The Molecular Level quest.
Stages of "From Within" quest
1. Locate Doctor Li
During my infiltration of the Institute, Elder Maxson has ordered me to make contact with a Doctor Madison Li and convince her to work on a "special project" for the Brotherhood of Steel.
She is in the Advanced Systems in The Institute so you have to start Institutionalized quest and accept Father's invitation to be able to speak with her.
2. Speak to Doctor Li in Advanced Systems
3. Retrieve evidence to convince Doctor Li
In order to ensure Doctor Li's trust, I've agreed 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.
The entrance to Fev Lab is in the BioScience Systems.
Follow the corridors until you reach main lab.
Take the Brian Virgil Personal Log holotape.
You will also find here a serum that Virgil requested you.
5. Report to Elder Maxson
I've provided Doctor Li with evidence that the Institute was lying about Doctor Virgil's "death." As a result, she's agreed to return to the Brotherhood of Steel.
or (when you lost an access to The Institute)
Due to my actions, I'm no longer welcome at the Institute. I was therefore unable to convince Doctor Li to return to the Brotherhood of Steel. I'll have to report the failure of my mission to Elder Maxson.
6. Quest completed
I've informed Elder Maxson that Doctor Li is on her way back to the airport to work on the Brotherhood of Steel's "special project."
or (when you lost an access to The Institute)
As a result of my actions, Doctor Li's assistance will be impossible. Elder Maxson will have to find someone else to work on the Brotherhood of Steel's "special project."
Next / new quest is Liberty Reprimed (Main Story, Brotherhood of Steel).