Here's my CS, much later than it should've been.

[hider=Children of Atom]
Name: Church of the Children of Atom

Flag: [url=https://static.wikia.nocookie.net/fallout/images/f/f5/CoA_Banner_1.svg/revision/latest?cb=20191127033756]This is one of several flags they use[/url]

Territory: Megaton (under Brotherhood rule), the Glowing Sea, and the western half of Mount Desert Island.

History: The Church of the Children of Atom was a religion worshiping the god Atom that sprung up in the Capital Wasteland around an unexploded atomic bomb, based on the belief that every atom was a universe and that nuclear fission resulted in the creation of new universes.  In the Capital Wasteland, the church prospered due to both the economic boon brought on by the publishing of the Wasteland Survival Guide and the increased stability brought on by the Brotherhood's rise to power.  Missionaries and faithful set out to find other holy irradiated sites, and one such expedition, sent at the behest of Confessor Cromwell, came to Mount Desert Island.

On the island, the Children of Atom were aided by a synthetic man named DiMA, and settled in what was once an old submarine base, irradiated by the submarine's leaking reactor.  Tensions arose, however, when a missionary was murdered in Far Harbor, a tragedy that would cause Confessor Martin to depart from the island.  In his place, a ruthless fanatic named Tektus took power, giving himself the title of High Confessor and eliminating anyone he considered might be disloyal to him, and preaching the destruction of both Far Harbor and Arcadia.

Eventually, DiMA chose to act, and had Tektus replaced with one of his synth followers, programmed to believe he was the real High Confessor, with the best traits of both Tektus and Martin.  The synth Tektus proceeded to make peace with the other factions on the island and rein in the fanaticism that his predecessor had encouraged, and peace on the island was achieve.

When the Institute survivors took Arcadia, the synth Tektus was upset at this shift in the balance of power, and wary at the new arrivals' much more liberal use of DiMA's fog condenser technology, but the two sides ultimately reached an understanding, where the Institute would control the eastern half of the Island and the Children of Atom would rule over the western half.

In practice, the Children of Atom only control the southwestern part of the island, preferring to leave the northwestern half of "Atom's kingdom" largely untouched and unsullied, aside from a few retaliatory strikes against trappers.  They've taken over the Fringe Cove Docks to use as their own harbor, and have taken Brook's Head Lighthouse from the trappers to enable their ships to safely navigate the fog.

Pressing Issues: Currently, the Church is seeking to re-establish contact with its other branches.  Last they've heard, the church in Megaton remains strong in the faith and prosperous, but due to being under the Brotherhood's rule, is politically and militarily powerless.  The situation in the Commonwealth is much more dire, with many its faithful being violent fanatics. and as a result the church's reputation in the Commonwealth is very bad.  Currently, the High Confessor is looking to send missionaries and inquisitors to the Commonwealth to correct the fanatics and their misguided beliefs, and to send messengers to Megaton to ensure that their brothers and sisters aren't being mistreated.

They're not entirely sure what to make of the Institute.  It's unclear if they can be trusted, or what their long-term goals are, but what is clear is that they have considerable technical expertise, and so far have been willing to coexist.  There might be considerable risk in sharing the Nucleus' tech left behind by DiMA but also considerable opportunity if the Institute's scientists and engineers could make the Vessel- and the base housing it-operational.

As for Far Harbor, they're still upset at their killing of their missionaries, but have agreed to peace, or at least mutual non-aggression.  Since the Institute now claims sovereignty over Far Harbor, this has not changed.  Most of the harbor's inhabitants are welcome to come to the Fringe Cove docks to trade, except for Allen Marks, whom the Zealots and Inquisitors have standing orders to arrest if he sets foot in Children of Atom territory.[/hider]