Determine which group(s) a running process is in?

The list of groups is given under Groups in /proc/<pid>/status; for example,

$ grep '^Groups' /proc/$$/status
Groups: 4 24 27 30 46 110 115 116 1000

The primary group is given under Gid:

$ grep '^Gid' /proc/$$/status
Gid:    1000    1000    1000    1000

ps is also capable of showing the groups of a process, as the other answers indicate.

For the effective group id, real group id and supplementary group ids (as used for access control):

ps -o gid,rgid,supgid -p "$pid"

gid and rgid are fairly portable, supgid less so (all 3 would be available with the ps from procps as typically found on Linux-based systems).

group, rgroup and supgrp can be used to translate group ids to group names, but note that for group ids that have several corresponding group names, only one of them will be shown (same as for ls -l vs ls -n or anything that deals with user or group names based on ids).

For the process group id (as used for terminal job control):

ps -o pgid -p "$pid"

To store it into a variable:

pgid=$(($(ps -o pgid= -p "$pid")))

Using ps:

$ ps -o group,supgrp $$
GROUP    SUPGRP
muru     adm,cdrom,sudo,dip,www-data,plugdev,lpadmin,mlocate,sambashare,lxd,libvirtd,docker,muru

From man ps, the output columns used for -o:

   egid        EGID      effective group ID number of the process as a
                         decimal integer.  (alias gid).

   egroup      EGROUP    effective group ID of the process.  This will be
                         the textual group ID, if it can be obtained and
                         the field width permits, or a decimal
                         representation otherwise.  (alias group).

   gid         GID       see egid.  (alias egid).

   group       GROUP     see egroup.  (alias egroup).

   supgid      SUPGID    group ids of supplementary groups, if any.  See
                         getgroups(2).

   supgrp      SUPGRP    group names of supplementary groups, if any.  See
                         getgroups(2).