If your grouped objects are closed curves, you can
- Duplicate the group,
- Use "path > combine"
- Put the combined path in the background
The combined path is the silhouette of your object. This silhouette can have the thick lines and fill color, while the other group would have no fill and the thinner lines. The outer lines would still be duplicate (with thinner on top of thicker), unless you erase them by hand (hold down Ctrl and click inside the group).