It is very awkward to have the Group Header on one page and then the details on the next. How can I keep them together (I have "Keep Together" checked on Group Header and Footer, as well as Details)?

"New Page [before/after]" is not checked.
 
Maybe I should explain further. I have groups with varying amounts of detail. There might be anywhere from 1 to 10 groups on a page. It's only when a group is near the bottom of the page, and there is room for the group Header but not the Details, or room for Header and Details but not the Footer, that it happens.

in your Report, go to the Menu - Report | Group Expert, now choose on the right pane the Group you are grouping by, and then clik on the Options button. that brings up the Change Group Options window, clik on the Options Tab and place a check mark in the Keep Group Together.
 
(several ways to access the group options) this is the option that will start your Group if it has details that won't fit on the page to start on the next page.
 
hth,