After you decide to Join the Institute faction, you will be assigned with few quest in order to gain Trust of this faction. In This Institute Quest Walkthrough, we will complete the first three quest: Synth Retention, The Battle of Bunker Hill and Mankind Redefined.
This is the very first quest after you have agreed to Father with you joining to Institute. A very simple and harmless quest with just 4 to 5 enemies.
Objective
Walkthrough
Your Mission will start after talking to Father, he will ask you to meet Courser to know more about the Rogue Synth. Then Courser X6-88 will be asked to assist you with the mission. You need to Leave the Institute and head straight to Nahant Sheriff's Department. Now that the department is deserted and heavily guarded with Raiders and Turrets you might want to avoid all the risk.
Either you try to swim to the Synth's location which is near the Shanty tower on a sunken tanker or check the platform from the Trash boat and use it to travel. Once you reach the marked location(Libertalia), climb up to enter the Captain's Cabin now first thing to do here is grab the Legendary shotgun Le Fusil Terribles and then interact with Courser X6-88 to get the reset Code for the Synth.
The Rogue Synth(Gabriel) will be along with his 2 Lieutenants so whether you can reset him with the code Courser gave or take him and his companion down it's all up to you. Once you have completed the Mission, return to Father and he will reward you with your own living quarter along with supplies. Once you are done checking the room go ahead and talk to Courser X6-88 to proceed with the next Quest.
This is the second Essential Quest in Institute faction where you need to choose which faction you are loyal with and with that the game scenario will also change.
Objective
Walkthrough
Firstly go and speak to Father where he will alert you with some stolen Synths ans ask you to get it back from the Bunker Hill, but you have the option to select the Faction you are Loyal with and here are the option and scenarios which may appear:
If you Choose to Stay Loyal with:
Note: Even if you choose to stand against the Institute, the next Quest Mankind Redefined will still be available in your activity log after this quest is over.
If we presume the you stand with Institute then continue with X4-18 to get the Synth recall codes, ready to fight against the Strongholds and finally get access to the Basement's Trapdoor. Now that you are done dealing with the enemies and turrets it's time to deal with the Faction loyalty and here is the option you will get to choose what to do with the Synths if you are with:
Leave the Bunker Hill and get back to the C.I.T. Ruins to the west after your done dealing with the Synths. You need to get inside the Ruins from the ground floor entrance. While you get to the marked location the Synth will have the Super Mutants positioned throughout the location, so get to the roof quickly either via the battleground or the stairwell and once you reach the top go and have a word with Father to complete the side Quest.
Another Essential Institute Mission, but a very Short one which start just after the Bunker Hill Mission. Here you need to do is Attend the Institute Conference Meeting.
Objective
Walkthrough
Once you had a word with Father at the Rooftop the next Essential Mission will begin called the Mankind Redefined. Now return to Institute and go straight to the conference room. Simply start a debate with the individuals and then agree that you still gonna help them with their cause.
Now X6-88 is available as the Companion and you can take him along at the Juncture from now on. With the Meeting, the Mission is also done and then go ahead and talk to Father again to continue with the other Essential Missions.