In the default role groups there is a group named "Recipient Management", which gives the permissions to manage mailboxes and groups.
To give a delegate this permission, you can add him to this role group in Exchange Admin Center:
If you need to do this in Active Directory, please open Server Management>Tools>Active Directory Users and Computers:
Locate Microsoft Exchange Security Groups>Recipient Management, then add the user as a member (it does the same thing as in Exchange Admin Center):
Besides, if you need more specific permissions, like only let the delegate be able to create mailboxes/groups but not allow him to delete mailboxes/groups, please share the detailed requirement so we can give suggestions based on your requirement.
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