Rename user's group name

I just did usermod -l login-name old-name to change the username, but the group name is still the old one.

Any way of renaming the group name? I can't find any info to do so.

groupmod should work fine:

$ groupmod --help
Usage: groupmod [options] GROUP

Options:
  -g, --gid GID                 change the group ID to GID
  -h, --help                    display this help message and exit
  -n, --new-name NEW_GROUP      change the name to NEW_GROUP
  -o, --non-unique              allow to use a duplicate (non-unique) GID
  -p, --password PASSWORD       change the password to this (encrypted)
                                PASSWORD

Use it like this:

groupmod --new-name NEW_GROUP_NAME OLD_GROUP_NAME

To reNAME as you asked:
sudo groupmod -n new-name current-name

Note: If the name is already in use but you want to use it anyway:
sudo groupmod -o -n new-name current-name

If you are root you don't need to use sudo