Applying to become an organizer of an existing group is easy. Simply navigate to the group's page and join if you are not already a member.
On the right hand side you'll see a link that says "You are a member of this group," just to the right of the "New Post" button:
Click on that link and then select "Apply to become an organizer.
Then follow the instructions! Contact the existing group organizer if you have any questions.