This is the expected behavior in your scenario, as detailed for example here: https://learn.microsoft.com/en-us/exchange/troubleshoot/groups-and-distribution-lists/cannot-manage-dg
The article also explains the available workarounds.
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We have a Hybrid setup. Our domain functional level is 2016.
The users mailbox are in EXO, but Distribution List/Groups are on-prem.
In AD, we have checked the box for "Managers can update membership list".
When the Manager open the address list in Outlook and try to modify members, he get an error message as soon as he try to save the change. The error message is:
"Changes to the public group membership cannot be saved. You do not have sufficient permission to perform this operation on this object."
Wen I look at this group, I see that this user is set as the Manager. So why is he unable to save the changes after he made some modifications?
We use the classic Outlook, not the new version.
This is the expected behavior in your scenario, as detailed for example here: https://learn.microsoft.com/en-us/exchange/troubleshoot/groups-and-distribution-lists/cannot-manage-dg
The article also explains the available workarounds.