After a recent consolidation we needed to give users in a seperate (but trusted) domain access to mailboxes on an exchange 2010 server.
The below code (copied from https://social.technet.microsoft.com/Forums/exchange/en-US/d2f051f0-9649-4ac0-8e82-9783a34d0eed/how-do-you-grant-full-mailbox-permissions-to-someone-in-a-different-domain-in-exchange-2010?forum=exchange2010 – by J. Duke Rogers Communicore Technologies & Triangle Forensics) does the job. I added the bit to get the Distinguished name from the mailbox (saves a lot of typing).
$mb= get-mailbox “mailbox_to_be_given_rights_on”
$dn = $mb.DistinguishedName
Add-ADPermission -Identity $dn -User ‘mydomain\mygroup’ -ExtendedRights ‘Send-as’
Add-MailboxPermission “mailbox_to_be_given_rights_on” -User “mydomain\mygroup” -AccessRights FullAccess
