Thanks, Frances.
Yes, you can. The photo is part of Admin, Settings - a separate button.
/g/GroupManagersForum/wiki/Quick-Start-for-Group-Formation#Cover-Photo-button3A
I've set all these, but the home page HTML still has:
<meta property="og:url" content="/g/zoom-rxx" />
<meta property="og:type" content="website" />
<meta property="og:title" content="zoom-rxx groups.io Group" />
<meta property="og:description" content="Discussion group for users ..." />
<meta property="og:image" content="/img/logofull.png" />
As you can see, the og:image refers to the generic groups.io logo, not my
group image.