You should be able to script that. As long as you target a group the script could duplicate the group into a new document with the document named for the group and parent and current date time. Save the new document to the current document path folder as a jpg and close the duplicated document.
My reason was because with one function you named the output file and isolated the Layers then save as and close three steps. No matter how you look at it you saving a new document. You can use different steps to save that new jpg document from the current document. Duplicate group seemed to be an easy clean way to do it without having to change anything in the current document.