With Defend The Castle complete, you and Preston want to take the fight directly to The Institute. This quest also assumes you completed the Inside Job quest during one of your prior visits to The Institute. Talk to Sturges to learn what he found out from the holotape.
He found a point of entry to The Institute, thanks to a sewer passage that leads to one of the 200 year-old facilities in the C.I.T. This entrance is not too hard to find, thanks to the waypoint indicator. It's slightly east of the C.I.T. ruins, along the river.
Follow the lengthy sewer path by swimming and by going on foot. Expect a handful of creatures and a couple Synths along the way. Somehow, you'll end up in The Institute, after you find and enter an open panel in one of the sewer tubes.
Now you're in the mass relay room. Activate the console to teleport The Minutemen, including Preston Garvey. After speaking with him, head to the reactor as indicated by the waypoint. Unsurprisingly, The Institute has locked down the Advanced Systems lab entrance leading to the reactor.
Take a detour up to Father's room and access the nearby terminal to override the lockdown. When you've finally made it inside the Advanced Systems lab, head toward the reactor chamber. Reach the reactor and activate it. Sturges will teleport you back to the relay room, after which you'll need to speak with Preston and rescue the Synth version of Shaun. Step into the relay once more. Activate the detonator to finish off The Institute, and then speak with Sturges to complete the quest.