When on-premises distribution groups are synced to an Office 365 tenant via Azure Active Directory Connect, migrated users who are owners of the distribution group can't manage them in Outlook. Depending on the version of Outlook used, the user will receive an error message that resembles the following:
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This happens because Outlook tries to update the same directory where the user's mailbox exists. If the mailbox is in Exchange Online this is the Exchange Online Directory Service (EXODS) directory, which syncs with Azure AD. Since EXODS is read-only in a hybrid environment using AAD Connect, the user receives the error.
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The action 'Update-DistributionGroupMember', 'Identity.Members', can't be performed on the object '<name>' because the object is being synchronized from your on-premises organization. This action should be performed on the object in your on-premises organization.
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Changes to the public group membership cannot be saved. You do not have sufficient permission to perform this operation on this object.
This happens because Outlook tries to update the same directory where the user's mailbox exists. If the mailbox is in Exchange Online this is the Exchange Online Directory Service (EXODS) directory, which syncs with Azure AD. Since EXODS is read-only in a hybrid environment using AAD Connect, the user receives the error.