Hi @ Vijay Kumar ,
1.Azure AD Connect doesn't automatically synchronize Send As permission between on-premises Exchange and Microsoft 365 or Office 365, so cross-premises Send As permissions aren't supported. However, you can manually add Send As permission in both environments, as described at this link:Permissions in Exchange hybrid deployments | Microsoft Learn
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2.If you enabled the archive mailbox before migration, the archive mailbox is migrated to the cloud together when the primary mailbox is migrated. If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
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