This is about adding a user to additional groups, not changing the user's default group.
The classic SysV command is
usermod -G group1[,group2...] username
It adds a user to the named group(s).
But also it deletes the user from all not named groups!
Proposals exist to first list the user's existing groups with commands like
groups username
id -a username || id username
And then use the usermod -G with the existing groups plus the new group(s).
Alternatively use a text editor and in /etc/group add the user to the desired groups.
But why should you do all this? It's a computer isn't it?
Let's have a look at the different OSes:
AIX and HP-UX
They refer to their "smart" admin tools smitty and sam, respectively.
Nothing for the command line.
Linux
The usermod got -a ("add" option) to simply add a user to a group:
usermod -aG group1[,group2...] username
But - there is no corresponding delete option!
Well, meanwhile some distros have got a
usermod -rG group1[,group2...] username
And there is the gpasswd command that was enhanced with "add":
gpasswd -a username group1
and "delete":
gpasswd -d username group1
Solaris
Finally there is a nice implementation (in Solaris 11?):
usermod -G [+-]group1[,group2...] username
A prefix + means "add to the group(s)", a - means "delete from the group(s)".