Controlling group ownership of book pages

I want to have books for groups. As long as child pages are created starting at the group page, everything is good -- the child pages belong to the group. But if the poster starts at mysite/book, strange things can happen.

If "Audience required" is optional and the poster does not check a group, the page becomes a public page. It's a child page in a book that belongs to a group, but the page itself is public -- if someone who is not a member of the group knows the URL, they can go to it, and the content is viewable from Recent Changes and is emailed to everyone by Notify.

If "Audience required" is required, the poster can chose a group different from the group that owns the book. This results in one group owning the book but one page of the book belonging to a different group.

How do I force the child pages to belong to the same group as the parent page?

I'm running Drupal 5.12, Organic Groups 5.x-7.3
I have enabled: OG, og_access_control

Organic groups access configuration:
- Visible only within the targeted groups.
- Group administrator chooses whether her group homepage and audience are private or not. Defaults to public.

Organic groups configuration:
- Group creator chooses whether her group appears in the directory.
- Group creator chooses whether her group appears on the registration form.
- New members are not subscribed to group email notifications by default. [I'm using the Notify module]
- "Audience checkboxes" is unchecked

I wonder if "Audience checkboxes" is working correctly. When a member of a group creates a child page, they see a check list of the groups they belong to.