mmm. OK. are you creating wordpress pages and then using a shortcode or similar to place a given group on a given page? Do you have a custom field on the pages to hold the ID of the group?
What I am thinking is that you could have a hidden field in the form which is the group (or group ID).
Then, in the child theme functions.php you can have some custom PHP that checks if a page is being loaded, and if it is, to find the value held in the custom field that holds the group ID, and if it does have a group ID, to then use our sf_edit_query_args filter to modify the group ID field in the form to be that of the custom field on that page.
The only problem that I can see is that S&F forms require a results url, which would be different for each page. I cannot see how you would get around this.