Plugging a Leak is an optional Institute side quest. You can pick this quest up from Justin Ayo in the Institute SRB. This quest is obtainable after completing Mankind Redefined. This quest is mutually exclusive with Railroad faction quests, so if you joined the Railroad by completing Tradecraft you won't be able to pick up this quest.
After picking up the quest from Justin Ayo in the Institute SRB, head to Institute Robotics. Here you can talk to Alan Binet to provide more information on the Synth escapes. Access a robotics terminal and choose the view access logs option. Once you have the necessary information, your next destination is the Binet Quarters.
Once at the Binet Quarters, talk to Liam Binet, then access the Binet Quarters Terminal., and choose the view access logs option.
Your next destination is back at Institute SRB. Once here, access an SRB Terminal and choose View access Logs. This will lead you to the next location which is remote maintenance terminal 6. Choosing the List of Informants option on this terminal will give you some information unrelated to the quest, but you may find very interesting.
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Follow your map marker towards the maintenance terminal 6 area. You'll find Eve here, and she will confess that the culprit is Liam. If you pass a speech check with her you can avoid killing her.
After dealing with Eve, return to Liam Binet. He'll ask you not to turn him in, and instead frame Justin Ayo.
If you choose to turn Liam in, return and talk to Justin Ayo, and you'll be rewarded with 1000 caps.
Alternatively, if you choose to frame Justin Ayo, be sure to choose the ask for a reward option first, otherwise you'll get no quest reward. Head to Justin Ayo's quarters, and access his terminal, and disable security safeguards on his terminal. Return to Liam to complete the quest, and you'll be rewarded with Liam's Glasses. (They provide +2 INT, so may be useful for gaining more experience and making terminal hacking easier).
