You can create actions sponsored by this group and emails and mobile messagesusing this group's list in two ways, much the same as group organizers can (except they don't have access to group emails or reports).
First, from your group's manage 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, call campaigns, fundraisers, reports, and emails) 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 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 other group administrators can edit your actions, emails, mobile messages, or reports when they are sponsored by the group, as administrators can manage any sponsored action, email, mobile message, or report.