Fallout 4 created by Bethesda Game Studios.
Images used for educational purposes only.

Prerequisites: Complete Synth Retention
Trigger: Speak to Father
Reward: Varies; see below

Once you’ve completed Synth Retention and checked out your new apartment at the Institute, speak with Father. He’s peeved about ‘stolen property’, and wants you to get it back for him. This will necessitate crossing the Railroad, and the Brotherhood of Steel won’t be too happy about it, either. As such, this quest can go in several different directions, depending on your affiliations and choices.

Regardless of which scenario you choose, you’ll be heading to Bunker Hill, located to the northeast of the C.I.T. Ruins. It’s a small settlement which you may have already visited as a member of the Railroad. Depending on how you proceed, though, you may make a pitstop or two first.

The Battle of Bunker Hill in Fallout 4. It would be more epic
if anyone shot at you.
Work for the Institute

In this scenario all you’re doing is completing your task as requested by Father. Travel to Bunker Hill, then head south a block. You’ll find a Courser named X4-18 standing between two buildings. Speak to him and he’ll explain the mission and give you Recall Codes for the Synths you’re trying to capture. (If you’re persuasive enough he’ll also hand over some Synth Relay Grenades.) This conversation will trigger a massive attack on Bunker Hill, with the Institute, the Railroad, and possibly the Brotherhood of Steel all participating. None of them will be shooting at you unless you’ve made an enemy of one of them, in which case… keep your head down.

The front gate of Bunker Hill is locked, so sweep around to the eastern wall and look for a side door. From here you can access a Utility Basement behind one of the vending stands. The battle rages even more fiercely down here, but still, no one will attack you. The only exceptions are a pair of Turrets along your path. Keep an eye out for ‘em. Other than that, it’s a straight (well, twisty-turvy, but straightforward) run to the room with the Synths.

Once you find the Synths you’ll have multiple choices for what you can do with ‘em:

Use the Recall Code on each of the Synths to deactivate them.
Set them free. This will cause X4-18 to become hostile, and you’ll have to kill him.
Kill them. This will turn X4-18 hostile, and, again, you’ll have to take him down. (Strangely, the game seems to treat this as though you let them go, or at least it did for me.)

After the quest, head to the Ruins of C.I.T. and access the roof via the CIT Rotunda. Look for stairs up in the east wing. (Not the first set of stairs, either. Go until you’re forced upstairs.) If you complete the quest to Father’s satisfaction, or if you apologize to Father for letting the Synths go, you’ll be given the quest Mankind - Redefined and remain an ally of the Institute. (You’ll also get 500 Bottlecaps and 500 experience for doing things exactly as asked. You only get the experience if things went wrong.) 

If you express anything other than contrition after refusing to deal with the Synths Father will consider you a traitor to the Institute cause. You’ll gain - and immediately complete - Banished from the Institute, which will prevent you from ever fast travelling into the grounds again. This bars you from any further Institute quests, and prevents you from gaining an ending related to allying with the Institute. Institute Synths will also become hostile on sight once again.

Inform the Railroad / Brotherhood of Steel

In this version of the quest you need to head to one of the two faction headquarters and speak to their respective leaders (Desdemona and Maxson). Both will ask you to perform a specific task; Desdemona wants the Synths rescued, which is done automatically by killing X4-18, while Maxson wants the Synths destroyed, which will necessitate X4’s destruction regardless. The overall results are the same when you speak to Father.

You can optionally choose to inform both sides. This ultimately doesn’t seem to have much of an effect on how the mission plays out, though it does help confuse which side you want to help. Yay?

(It should be noted that the Brotherhood often just shows up on its own, whether you inform them about what’s going down or not. I have no idea why this happens.)