right click the group, select Change Group/Options tab put a check in Repeat Group Header On Each Page. This won't move the group header to a new page, it will just repeat it.
It can get tricky, and I have never had good luck with it, so I don't use this unless I absolutely have to, would be count the rows printed on a page and if your row count exceeds a certain number to have the group header perform a new page before (option in the section expert). Maybe instead of row counting you add size of each section...will have to remember this next time something like this comes up.
Keep group together is a cleaner answer in my opinion, but what the customer wants trumps my wishes for a nice report.