Email groups in a Hybrid Exchange environment

We have just completed most of our email migration to Office 365 from Exchange 2013 on-premise. We currently still have an Exchange 2013 server on premise, but that may not always be the case. After migrating, I found that newly migrated group owners could no longer change email group membership via the Outlook address book. After researching the issue, I found that this is a known issue and for groups created on premise, the ownership rights are not translated to O365. I have found no way to correct this. I then create a group in O365 (which does not get the address policy I designated on premise) and it is not seen by on premise users and Applications that send out SMTP email (this is because the On cloud entries are not replicated down to the local AD - DirSync only goes one way).

So I find myself in a problem where O365 staff cannot update email groups via Outlook (for those groups created on Premise OR On premise staff and SMTP Applications located on Premise cannot use groups created in the Cloud.

Can anyone verify if I this is correct - that I can only have groups one way or the other? If not, what are my options?
I do know I can continue to create groups on premise and direct users to change memberships with AD tools, but I am hoping there are some other ways to get this to work without relying on the On Premise Exchange servers which may go away. I know some deployments of O365 that are only in the cloud and do not see how MS can leave all current groups to have no owners.

Thank you,
John WaterAsked:
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Vasil Michev (MVP)Commented:
You pretty much got it right. The usual advice is to recreate groups in the cloud if you are planning to remove the on-prem servers. You might have to adjust the delivery settings, so they accept mail from outside the organization.

Groups write-back to on-prem feature is coming, but it will require Azure AD premium licenses.

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John WaterAuthor Commented:
thank you for the prompt response
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