That doesn't sound like expected behavior - it could possibly be a bug. One thing I would try, though, is setting the default sort on your group.
1. Right-click on your grouped field in the data pane.
2. Select 'Default Properties'.
3. Select 'Sort.'
4. Use the Manual option to get the sort the way you want it.
Another possibility would be to set the fiscal year start for the date.
1. Right click on your date field (not the grouping of the date field) and select Default Properties
2. Select 'Fiscal Year Start.' Set to July.
If these don't work, either post a packaged workbook to the forums or open a support case.
Did what you said to do, saved and then closed. When I reopened it went right back to the alphabetical sort. I did a group on appt date. I do have Fiscal Year set to July already. After I did the grouping of month and year, I renamed their group. Could this be why?
Without a look at your packaged workbook, I don't know that I'm able to give any more guidance. I tried mocking up a version of this with some fake data based on what you've written here, and my sort on the grouped dates persisted after I saved and closed the workbook, then re-opened it. It seems I am likely following different steps than you are, and there are just too many variables for me to be able to make an accurate guess without even a screenshot of your canvas. If you can post your twbx here, or a version of what you're doing with fake data, I can take a closer look.