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How to Manage Distribution Groups from Office 365 in a Hybrid Environment

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:
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.
-or-
Changes to the public group membership cannot be saved. You do not have sufficient permission