What do you mean by a "custom logo".. it has a custom avatar defined, along with a custom banner... as shown in the post you quoted.
BTW, the above post I am replying to is rather screwed up. It allows ME to edit it... but it appears that you edited my original post and put a reply into it instead of actually replying to it?
I don't know if it's an add-on you have enabled here... but there are some serious issues if so. In fact, the below links below your post allow me to use ANY of those links.
Which is the header of the message (notice the ability to "report, like, quote, reply"... then at the bottom of the message I see this
But yes, there is a custom avatar created for that specific group (as was shown), a custom profile banner (as was shown), and the logo_og is correct and picked up by FB and Twitter when shared (as was shown)... the image that is used when sharing that group IS NOT any of those. It's simply an image that was used as a notice avatar.