Add a user group programmatically

Hello,
Is there a way to add a group programmatically? Which action/filter do I need to use?

I have a membership site that requires the group to register (and pay a fee) before group members can register (and also pay a fee). I’m using the Simple Membership plugin to deal with payments and membership types.

As part of the post-payment processing, I’d like to automatically add the group so that when the group members register, their group is present on the registration form.

Any help would be appreciated.

Joe.