Hi IQA,
Normal mail-enabled distribution group and security groups can be synchronized to Office 365. However, dynamic distribution groups cannot be synchronized to Office 365 currently. We need to create them in Office 365 manually.
As for the issue about receiving emails sent to a group, I would like to collect the following detailed information:
1. Did the issue occur after a local user or Office 365 user sent emails to the affected distribution group?
2. Could you please check the distribution group membership on the local or Office 365 side?
3. Could the affected sender send emails to the newly created Office 365 users directly?
Thanks,
Claud