No Linux (não no BusyBox), Solaris, NetBSD, OpenBSD:
usermod -g group
The usermod command modifies the system account files to reflect the changes that are specified on the command line
-g, --gid GROUP
The group name or number of the users new initial login group. The group must exist. Any file from the users home directory owned by the previous primary group of the user will be owned by this new group.
The group ownership of files outside of the users home directory must be fixed manually.
No FreeBSD:
pw usermod -g group
No BusyBox:
addgroup -g user group
Você não poderá fazer isso sem privilégios de root.