If you are an owner of a group, you are able to edit your group. When you are on a group homepage, click on Group Options drop down. Select Group Settings to edit your group.There you can change the title, description, category, url, group type and your icon. Don't forget to hit save once you have made your changes.

If you want to learn more about creating a group check out this post and if you want to add customized widgets to your group check out this post.