Background
In Active Directory, the primary group is stored differently than normal group memberships.
Normal group assignment
Membership in the group is stored in the "member" field of the group.
Primary Group
Here, the primary group is determined directly in the user object via the "primaryGroupID" field. However, this field does not contain the group, but only its RID (Relative Identifier).
This means that the assignment of a primary group is not stored in the "member" field of the respective group. The group itself does not "know" that these members also belong to it!