This is what's been in my head:
A man from Groam would be a Groaman. Several men from Groam would be Groamen.
A person or entity belonging to Groam would also be Groaman, although Groamish or simply Groam are also acceptable adjectives.
So the way I've been thinking about it: The Groam Council, the Groaman Council and the Groamish Council are all correct, although one form might be more common in a particular dialect than others.