Where is the owners mailbox?
There is where you set the send as. I am assuming you are syncing the group from on-prem to 365.
If its in 365, then:
if the owners mailbox is on-prem
Exchange Server 2016 in hybrd with O365. Trying to set up an owner of a distribution group to be able to Send As (which I thought they'd have as an owner but they're getting the "You don't have permission to send on behalf of the specified user). Go into the group, UI open up to the settings > group delegation, + to add user. The only users in the list are the other distribution groups. I don't see a list of users.
I have not tried any PS commands yet. We've been in this set up for years and I can't imagine they haven't tried this before, so I'm concerned something changed.
I just found this article, wondering if this could be helpful.