To upload your group's avatar you must first be the creator of your own group.  To learn how, click here.

Go to your group's homepage.  On the upper right hand side, click on the "Edit Avatar" tab.  Here you will be able to browse your hard drive and upload your own picture for the group's homepage.  Once you click on the "Upload" button, it will automatically set your picture as your avatar.  To change it, just repeat these steps.