Aground

There are apparently several ways you can do this. This is just the route I took. After finishing the quest with the scavenger for the drones on earth (see my solution if you need help with that one) he will task you to find the professor and give you a keycard to his door. Fly around until you see a large crashed ship similar to the one west of starting island. You will see some mine tracks going down the side of a small ice block hill east of the wreckage. Follow these tracks to his door and open it with the card.
Complete his quests, including building the mobile lab, and eventually he will give you a quest to make an alien locator. The lab he is standing in on the far right of the mobile lab lets you research stuff for the cost of “time”. You research much faster if you add more survivors.
Once you get the locator researched it will find the alien home world and mark it on your galaxy map. If you fly there in the mobile lab, a cut scene will start, showing you being rescued by the colony ship and then you will automatically warp away to their home world, unlocking the achievement.
Can’t personally confirm the other methods, but if people post in the comments I’ll try to update.

If anything is unclear, please let me know.