You become a mod by increasing your Activity Score above approximately 50 if there are no mods; above approximately 70 if there already is a mod
NOTE: it slowly increases over time otherwise eventually everyone would become a mod. You may need more than 50 or 70 points if the group has existed for a long time).
Once you do, you'll see a "become mod" link at the top of your group's page. Click that, and now you're a mod.
Groups are limited to a maximum of 3 mods, unless you are a Supporter where you can promote as many as you'd like!