The Puppet Masters is a short main quest associated with Halcyon. You have to go to Byzantium. This page of the guide to The Outer Worlds has a walkthrough for this quest.
List of objectives
- Meet Sophia Akande.
The quest is activated as soon as you finish Balance Due. As a reward for solving the problem, Udom Bedford will hand you the navkey to Byzantium. He will also inform you that Sophia Akande is waiting to meet you.
You can go to the Unreliable and travel to Byzantium.
After landing in Byzantium, you immediately go to The Board office. There, after a short briefing, you will be allowed inside. Take the elevator upstairs and talk to Sophia Akande.