Warning:
This is where the story will branch off so if you want to try all possible questlines for every faction without starting a new game, I suggest creating a back up save now.
After getting the courser chip, you'll have to find someone who can decode it. Here are the possible options:
After the chip is decoded, it will be added to your inventory. Next, you have to return to Virgil to receive the receive the schematics for the teleportation device. You'll have three options: ask the Minutemen, Brotherhood, or the Railroad. You'll have to talk to the following persons, depending on the faction you want to ask help with.
Minutemen – Sturges
Brotherhood – Proctor Ingram
Railroad – Tinker Tom
Warning: If you haven't allied yourself with the Railroad yet and if you're still planning to, make sure to find them first because entering The Institute. If you entered the Institute before joining them, they'll permanently lose their trust towards you and will prevent you from joining the Railroad or doing their quests. That also means you'll lose the chance to unlock the ability to craft Ballistic Weaved clothing.
If you chose the Brotherhood, the teleporter will be built in Boston Airport. For the others, you can build it on any of your settlement, as long as there's more than enough space to do it. After talking to the engineer of your choice, you have to build four major components: stabilized reflector platform, molecular beam emitter, console, and relay dish. All the components can be built from the Special section of your crafting menu.
The assembled device is for one-use only. After the quest, it will overload and will just appear as normal junk. In the other hand, the generators you used to Power the teleportation device can be relocated and used to add power to your settlement's overall power grid.
The chosen engineer will move to the site where you're building the teleporter. After building the necessary components, assemble them then Power them up. You need a total of 27 power to enable it. Once you have prepared the necessary (dedicated power source recommended), wire all the components then talk to your engineer. When you're ready step in the teleporter platform to reach The Institute.
The quest will be completed as soon as you reach The Institute. The next quest however will depend on where you'll surrender the network scanner holotape you used within the Institute
Minutemen – Inside Job
Railroad – Underground Undercover
Brotherhood - From Within