Plugging a Leak

Plugging a Leak is an Institute side quest in Fallout 4. It can be accomplished for the benefit of the Institute or against their interests.

Detailed walkthrough
Speak to Doctor Justin Ayo at the Synth Retention Bureau. He suspects that someone within The Institute is helping Synths escape. His prime suspect is Doctor Alan Binet in the Robotics department. Go there and access any terminal (speaking to Binet is optional) and read the logs.

This will lead you to the Binet residence, where you will find the doctor's son, Liam Binet, and Liam's "mother", Eve. While there, access the terminal and read the logs. Go back to the Synth Retention Bureau, access one of those terminals and read the logs.

You'll be directed to the maintenance room, where Eve will be found acting suspiciously. She claims to have developed a motherly attachment to Liam and has been covering his tracks. Unhappy at being found out, she'll point a gun at you. At this point, either eliminate her or pass a hard Charisma check to convince her to stand down.

Whatever you decide, head back to Liam at the Binet residence. When you confront him, you'll have two choices:  Report him to Doctor Ayo or Frame Doctor Ayo.

Option 1. Report Liam to Doctor Ayo
Return to Dr. Ayo and inform him of Liam's actions to receive a reward of 1,000 caps and end the quest.

Option 2. Frame Doctor Ayo
Go to to Ayo's quarters and access his private terminal. Disable the safeguards and return to Liam to end the quest and receive Liam's glasses as a reward.

Bugs

 * Even if the option to frame Doctor Ayo is chosen, he will continue his duties as the acting director of the Synth Retention Bureau with no reaction to player actions or indication of when he will give up his position. Still, other Institute NPCs will reference how the player stopped Ayo from leading them on, as implied in Liam's cover story.
 * Terminal at the Synth Retention Bureau may not direct you to the maintenance room.