The list of groups a user is in is only set at login (and possibly modified by using the rarely used
sg command, but that only works if there is a group password set for that group). If that user logs out, then logs back in, they should then see themselves in that group. However, being in the group with GID 0 ('root' on most Linux systems, 'wheel' or 'adm' on most non-Linux systems) is not the same as having administrative privileges, and is in fact a bad idea from a safety perspective.
Why is this not the same as administrative priveleges?
The root user (UID 0) is special. The root group (GID 0, called
wheel on many non-Linux systems for historical reasons I cannot seem to find) isn't. Your user will be able to access files owned by the root group, but probably not much else. This is a somewhat complicated distinction, but is a very important one.
Why is this a bad idea?
This is a bad idea for pretty much the same reason that running as the root user all the time is a bad idea. You're always one mistake away from making the system unusable (on most Linux systems for example, being part of the root group means you can trivially make the system unbootable because you can delete things out of
What's the correct (and safe) way to give this user administrative privileges?
If possible, set up
doas or whatever other privilege elevation tool other than
su is present on your system to allow them access. If you have no such tool other than
su, install one (I would suggest
sudo, it's the most widely used and therefore one of the easiest to learn because there's lots of documentation). If that too is not an option, then just elevating through
su when you need to do admin tasks is the correct option (although do be careful doing so, as
su does not properly reset the environment for commands called through it).