Use of the apostrophe when possession is not the issue

What would be the best way to write "Beginners group" or "Childrens group" in "I'd like to teach the Beginners Group"? Both feel as if they need an apostrophe before the final 's', but this is not possession: the group does not belong to the beginners or to the children, the group is composed of beginners or children. Any ideas?

The possessive can be used to describe a relationship of composition. In a certain sense, the components of a thing do own that thing. It's analogous to indicating ownership with of:

• That's the home of Alice; that's Alice's home
• That's the group of children; that's the children's group.

(In saying this, I am thinking of a line from a German hymn: Welt ade, ich bin dein müde. Word for word, this means "World, adieu, I am your tired." Of course, the idiomatic translation is "... I am tired of you.")

For the beginners' group, the apostrophe comes after the s, because it is a plural possessive. While children's group is also a plural possessive, the apostrophe comes before the s because children does not end in s.

Another way to look at it: If you want to point out a group of which you are a member, you could certainly say "There's my group."