For example the group is called Bike Club
The newsfee will parse it like that: Bike Club - group at Oxwall Community Software Demo
And the image will be the image of the group, you can check it on our demo, besides, before hitting send button, you can hover over the parsed image, you'll get 'change image' button, where you can select the images provided on the groups page
If so, I just went there, the newsfeed parses the link strangely as a tooltip, besides you have video upload, link add and smile buttons below the status area, please disable all 3rd party plugins and let me know I'll do the testing again.
You will need to do that modification on your own or hire a programmer. You can do that here: http://www.oxwall.org/market/specialists
Besided I have passed this issue to our devs it will be fixed in one of the upcoming udpates. No exact date for now.