There are a lot of special achievements in "Frost Age 2", and self-exile is one of them. To achieve this achievement, you only need to make the exiled faction voluntarily leave New London. This kind of thing is relatively Trouble, the Windward Swamp colony needs to be expanded to meet the requirements of all the Exiles.
How to obtain the self-exile achievement in Frostpunk 2
Self-exile
Let the exiled faction voluntarily leave New London
Location: Story-Chapter 5
Colonize a new location for the faction and upgrade it with:
Housing
Food
Fuel (heating)
Once this happens, the opposition faction will no longer resist and leave
Expand the Windward Swamp colony to meet the requirements of all Exiles and then hand it over to them. Do not arrest or detain any rebels for deportation; if you do, the trophy will not be unlocked.