skinparam backgroundColor can set the background color for the image as a whole *and* for the boxes created by the "group" command. However, "transparent" works only for the image, and if you use transparent for the image, boxes are always #ffffff. There doesn't seem to be a separate color setting for the backgrounds of groups. The other thing I thought of was to specify a color first, then use another skinparam to select "transparent", but that just reverts to boxes being white.
I may be missing something obvious.