The brotherhood wants us to recruit a former ally that now works with the Institute… Dr. Li. If you played Fallout 3 you may remember the project that she worked on back then.
Dr. Li isn’t exactly happy with the secrecy of the Institute, but she’ll need some real reason to leave… thankfully we’ve already worked with Virgil, so we can tell her first-hand of the nasty things the faction has done previously.
Virgil’s holotape works as proof here, which I had stored in my home at Sanctuary. The holotape is discovered in a hidden lab, within the Institute. For help on that, check out the video “Virgil’s cure”.