ERROR: The action 'Set-DistributionGroup', 'AcceptMessagesOnlyFromSendersOrMembers', can't be perfor

Gregory2190 Yes absolutely, I should have posted this originally, my bad!

You must insert the full Distinguished Name of the Distribution Group in-between the quotes, and insert a users distinguished name that should be allowed to send to the group in the "authorig" quotes. For example:

Set-ADObject "cn=MyDistributionGroup,OU=Groups,DC=local,DC=com" -replace @{authorig="CN=Bauman\, Steve,OU=Users,OU=User Accounts,DC=local,DC=com"}

You can find your distribution groups distinguished name by opening up Active Directory Users & Computers on your AD server, enabling "Advanced Features" (inside of the "View" top header tab), then right-clicking the object, selecting "Properties", clicking the tab "Attribute Editor", and then scrolling to find the 'distinguishedname' attribute.

Let me know if you'd like some screenshots!

EDIT

To elaborate, if you're getting the error "Cannot find an object with identity", then the Distinguished Name was not typed in properly and the query was not able to locate the object in your Active Directory. Be sure to escape commas inside of each Distinguished Name component (as shown in the example above) if your object has commas in its Common Name.

The Active Directory Users & Computers Attribute Editor tab will not allow you to edit the "authorig" attribute if there are no entries inside of it (I have no idea why honestly). But once you add at least one Distinguished Name inside of it, you can use the Attribute Editor.

aballardAGI 

Yes I believe so. You can test it yourself by creating a test distribution group, setting this property, allowing it to synchronize to Office 365, then sending an email to it.

Unfortunately, deployment of this was halted internally for the company I work for, so I can't tell you for sure.