When a new group is created, the system assigns to it a default avatar. After that, group administrators can change the group's avatar using the Group profile web part.
Users can delete the avatar, uploada custom one, or select a predefined avatar.
You can find a live example of this on the Community Starter site. Sign in as a group administrator (e.g. Josh with blank password, Josh is the Australian travellers group admin) and click Groups in the main menu. You should see the Australian travellers group in the Your groups section. Click it and then use the Manage the group link in the Shortcuts section on the right.
Changing group avatars in the administration interface
Site administrators can change the avatar of any group. To change a group's avatar, open the Groups application and edit this group on the General tab.