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Based on your description, you are experiencing issues with group mail moderation in a hybrid environment. Here are a few steps you can take to troubleshoot and potentially resolve the issue:
- Make sure the Hybrid Configuration Wizard (HCW) has run and completed successfully. This is critical for proper mail flow between the on-premises and Microsoft 365 environments.
- Audit emails are processed through arbitration mailboxes. Verify that these mailboxes are healthy and configured correctly. You can check their status using the Get-Mailbox -Arbitration command.
- Even though you mentioned that the transport rule is not blocking the notification, it is worth revisiting the connector. Make sure the connector is configured correctly to route the audit email to Microsoft 365.
- Use the message tracking log to track the path of the audit email. This can help determine where the email is stuck. You can use the Get-MessageTrackingLog command for this purpose.
- Make sure the permissions required for cross-forest auditing are set correctly. This includes verifying that the Microsoft 365 auditor has the necessary permissions to approve or reject the message.
Please feel free to contact me for any updates. And if this helps, don't forget to mark it as an answer.
Best,
Jake Zhang