A group can run on its own address instead of its address under meetagain.org. Getting there needs two things that are not in your hands at the same time: DNS pointing at the server, and a platform administrator approving the change.
The domain screen is reached from the Custom Domain row on the Group tab. It shows the current domain, the requested one, and two requirement lines that have to both turn green before the change can be applied. The DNS check runs automatically every five hours, so most of the waiting is not something you can hurry along.
Request the change
Type the new domain into New Domain and press Request Change. Nothing moves yet; the request is recorded and the DNS check is scheduled.
Create the A record
At your domain registrar, point the domain at the IP address the page shows. The instructions block on the screen names it.
Wait for the DNS check
Propagation can take 24 to 48 hours, and the platform re-checks every five hours. The requirement line shows when it was last checked.
Wait for the administrator
A platform administrator approves the request - before or after DNS passes, whichever comes first. This is a manual step by a person, not a timer: the same person also sets the server up for your domain, including its HTTPS certificate.
Apply it
With both requirements met, the Change button becomes usable and the group moves to the new address whenever you press it.
The old address stops being the group's home
A domain change moves the group; it does not run it in two places. Anything you have pointed at the old address - printed flyers, links in other people's posts, a bookmarked page - is worth thinking about before you press Change, not after.