When you migrate a user, you can also have Directory Sync Pro for Active Directory migrate any groups in which they are a directly a member. Just check the Synchronize Accompanying Groups box. If they are member of a group, that is itself a member of another group, you can choose to include the entire nested chain of groups by checking the Include Parent Groups box also.
However, the group objects themselves must also be included within the scope of this profile, or another profile, or this setting will have no immediate effect. If you later fixed the issue by including those particular groups within a profile, the user's membership would then be included automatically at that time.
To avoid this issue, if you plan to use this option, you may want to first make a profile that includes all the groups, and only groups, in the entire domain. You would then run only a push synchronization, to get these items into the SQL database without pulling them into the target. Then this option would always behave as expected.