This is a confusing side quest, not because it's long or drawn-out (or even difficult), but because there's a lot of variations depending on the events that took place when you first escaped from Vault 101. As a result, we've written what we did during this optional side quest, but you might find things differently here. For instance, the old Overseer will be the one wreaking havoc on Vault 101's residents, but if you killed him when you escape, a story arc will be included where a new overseer is installed before you arrive. So, keep all of this in mind going forward, for some of the characters we ran into and events we took part in may differ from your own experience.
As long as you listened to the entirety of Amata's distress signal over the radio, you will automatically know that the password into the vault is none other than her own name (this will be inserted automatically). Once you're through the outer door, head into the next door, and you'll be intercepted by a Vault 101 guard (for us, it was Officer Gomez). He'll be friendly with you, even if you're somewhat hostile towards him, and ultimately tell you the situation in Vault 101. It appears people know that both your character and your character's father have escaped, and they want to see things outside on their own. But the Overseer won't have any of it. After picking Gomez's brain for everything he knows, don't let him go just yet. First, ask him to bring you to Amata. (Alternately, you can go see the Overseer immediately, though we opted not to during our particular playthrough).
Follow Gomez after asking him to bring you to Amata (if you asked him to bring you to the Overseer or opted to do some exploration on your own, then you are indeed on your own). You won't run into her immediately, however. First, you'll run into a fray between one of the Vault 101 rebels (those who want to leave the Vault) and a security guard named Officer Taylor. The rebel in question is Gomez's son (who you can talk to later), and although the trigger-happy Taylor fires a round at him, he misses. From there, you can speak to Taylor and then continue along towards Amata's location, which is in your father's old clinic. Remember -- you can stop following Gomez at any time if you want to talk to someone you encounter or explore an area. He'll wait for you until you're ready.
When you finally reach Amata's location in the clinic, you'll finally be able to get to the bottom of just what's going on. Amata fills you in on what's happened since you've left; things have certainly not been good. The Overseer is trying to keep everyone in Vault 101 indefinitely, even though the evidence is now clear that the Capital Wasteland outside isn't completely heinous to live in. People have died and subsequent lines have been drawn in the sand. It's up to you to solve this problem one way or another (though Amata begs you not to actually kill the Overseer -- what you do to him is up to you, of course).
Here's how we handled the situation. Remember! You don't have to do what we did. There are other options and other people you can work with here. We chose to work with Amata and put her at the helm of Vault 101, however. After all, we knew we could trust her! Navigate your way to the server area of the vault (marked as "Admin"). Here, you'll run into a man named Officer Wilkins. There's no reasoning with him; he'll go at you when you speak with him. Down him and then proceed. Thereafter, we ran into the new overseer, a man named Mack. We killed him too. Keep in mind that you may run into the old Overseer if you didn't kill him at the beginning of the game.
Once the deed is done, return to Amata. She won't exactly be thrilled that you killed the Overseer in cold blood, but she'll still talk the helm of Vault 101 as its new Overseer, since that appears to be what the remaining living residents want. Amata has every intention of keeping Vault 101 as their home, but they want to explore the outside world, trade with people out there, and learn more about the world outside. But, because of the havoc you caused earlier and just caused now, you're asked to leave… for good. Do so, and you'll successfully finish this side quest.
