When a new group is created, the default avatar will be assigned to it. After that, group administrators can change the group's avatar using the Group profile web part.
Users can Delete () the avatar or Upload a custom one from a file. If pre-defined avatars are enabled in site settings, users can also click the Select pre-defined avatar link, which displays a gallery of pre-defined avatars.
You can find a live example of this on the Community starter site. Sign-in as some group administrator (e.g. Josh with blank password, he is the Australian travellers group admin) and click Groups in the main menu. You should see the Australian travellers group in the My groups section. Click it, and what you can see in the screenshot below will be displayed to you.