The other day I created about 20 SharePoint Groups for a page that that was to show a view customized to each department head.
Another Site Owner who had full control on the site, was to test it for me but found that he could not add himself into the groups that I had made for the audience mechanic.
The issue is this, ONLY the Group owner, which was me, can add members. Site Collection Admins can of course, but site owners do NOT get rights to add members to any group by default.
The workaround is to make the owner of the group a SharePoint Group that includes all your site owners.