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Hi @Safs-3080
Please run the command below to check the permission for the shared mailbox
Get-MailboxPermission -Identity ******@XX.com | Format-List
If you see "isinherited" is false, it will go through. However there will still be instances where middle of the migration some new mailboxes will be created. For that, run a script at cloud to add those to migrated mailboxes.
According to the official document we can know that, Mailbox permissions migration: On-premises mailbox permissions such as Send As, Full Access, Send on Behalf, and folder permissions, that are explicitly applied on the mailbox are migrated to Exchange Online. Inherited (non-explicit) mailbox permissions and permissions granted to objects that aren't mail enabled in Exchange Online are not migrated. You should ensure all permissions are explicitly granted and all objects are mail enabled prior to migration. Therefore, you have to plan for configuring these permissions in Exchange Online if applicable for your organization. In the case of Send As permissions, if the user and the resource attempting to be sent as aren't moved at the same time, you'll need to explicitly add the Send As permission in Exchange Online using the Add-RecipientPermission cmdlet.
So re-apply the permissions after the migration will be a resolution.
Below is also an article introduces about the similar topic for your reference as well: Migrated permissions of migrated mailboxes in Exchange Hybrid based environment
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