Plugging a Leak

Plugging a Leak is a quest in Fallout 4.

Quick walkthrough

 * 1) Speak to Doctor Ayo.
 * 2) Access lab terminal.
 * 3) Go to Binet residence and access terminal.
 * 4) Confront Liam's mother in maintenance room.
 * 5) Confront Liam.
 * 6) Report Liam or frame Ayo.
 * 7) Receive reward.

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.

Bugs

 * PC players who joined the Railroad, sided with The Institute and want to play this quest can type setstage "000a8257" 1 into the console and then can approach Justin Ayo to receive it. Of course, this is a risky action and may cause problems if you also have, or plan to do, the Underground Undercover quest. It may also cause other bugs so it is probably safest to trigger it only once The Institute's quest-line is over.
 * 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.