I ran a report and when I select to add the results to a group I am not able to find the group I want.
System created group types like Event Attendees, Advocacy groups, TeamRaiser Participants, etc. are automatically populated with members based on the user interactions. Groups that are system created have specific process of adding members. If you would like members to be in one of the groups then an alternative would be to create a new group and set it to be built by query results that include one of the system generated groups as a query requirement.