No TestFlight invite email gets sent for internal users
For new builds on the same version, once the build becomes available, I had to turn OFF testing for that version and immediately turn it back on. This instantly sent an Update notification to TestFlightApp, emailed the testers, and allowed users install the new build.
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I've been round and round on this as well. What finally worked for me was to clear the internal tester status on the user, double check that the account was removed from the Internal / External list and then re-enable tester status for the user. Invite as an internal tester then worked for that user.