Plugging a Leak

Plugging a Leak is a quest in Fallout 4.

Detailed walkthrough
Speak to Doctor Justin Ayo at the Synth Retention Bureau. He suspects that someone within The Institute is helping Synths escape. That someone is Doctor Alan Binet in the Robotics Laboratory. Go the lab 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. 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 Liam's mother will be found acting suspiciously. Unhappy at being found out, she'll point a gun at you. At this point, either eliminate her or 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 recieve 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 recieve Liam's glasses as a reward.