when switching between itemgroup pages via the page buttons, the game always selects the top-left itemgroup of that page.
There is only one exception to this: Having the Search itemgroup selected.

In that case, going to the previous (vanilla) page, the search group stays selected.
However, when going to the next (modded) page from the vanilla page, it switches back to the top-left itemgroup of that page.

tested on fabric-item-groups-v0, version 0.1.0+1e69a0a542

This was entirely re-written since this issue was opened. Please open a new issue if this is still a problem in 1.19.3.

This behaviour also persists with the Saved Toolbars and Survival Inventory tabs.

I also don't think changing pages should change the selected group at all. For example, if I have decorations selected and I go to the next page, I think decorations should still be selected but I should be able to see the tabs from the next page.

I also don't think changing pages should change the selected group at all.

well I think it should, but only if the selected tab is not part of the next page.
(i.e. if the group is not inventory, search or hotbars)