Android offers seamless updates for apps from third party stores targeting API 29 since Android 12 via the REQUEST_INSTALL_PACKAGES permission:
https://support.google.com/googleplay/android-developer/answ...
https://developer.android.com/reference/android/content/pm/P...
There are issues on GitLab covering this:
https://gitlab.com/fdroid/fdroidclient/-/issues/1836