There are two ways to create actions sponsored by a group to add activists to the group's email list. First, from your group's participate page click the action type you want to create from the right sidebar under "Create A Sponsored Action."
Note that there are two additional action types available to groups that you can't get from the "Start Organizing" menu at the top of the page -- event campaigns and campaign pages. The only way to create these two action types is from these sidebar links.
You can also create normal action types (petitions, events, forms, and fundraisers) from the "Start Organizing" menu. As you are creating an action, you can choose the group to sponsor that action from the "Sponsored By" drop down menu. Your choices will be any group you are an organizer in or administrator of.
Note that as an action creator, you can remove yourself and only have the action sponsored by a group if you'd like. Doing this means your personal account will not get the data on activists who take action on this page, only the group will get the data.
Also note that group administrators can edit your actions just as you can when they are sponsored by the group.