The window of opportunity to pursue Plugging a Leak is a tight one, especially if you're the type who likes to please multiple factions. You will need to progress through the main Institute missions all the way to Mankind Redefined. In addition, this quest will be inaccessible if you complete the Tradecraft Railroad quest.
Speak to Doctor Ayo at the Synth Retention Bureau. He suspects that someone within The Institute is helping Synths escape. That someone is Doctor 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, and Liam's mother. While you're there, access their terminal and read the logs. Then go back to the Synth Retention Bureau, access one of those terminals and read the logs.
You'll then need to reach the maintenance room, where you'll find Liam's mother acting suspicious. Unhappy that you've found her, she'll point a gun at you. You can 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. To pull off the latter, go to to Ayo's quarters and access his private terminal. Then disable the safeguards and return to Liam. We recommend reporting Liam to Ayo, since the reward is better: 1,000 bottlecaps versus Liam's glasses.