Using the Same Alias in Multiple Domains in Office365

1. Start PowerShell as an Administrator
2. If you haven't before issue the command issue it now: Set-ExecutionPolicy RemoteSigned
3. Connect PowerShell to Office365 - (from http://technet.microsoft.com/en-us/library/jj984289(v=exchg.150).aspx)
1. Issue the command and input your credentials: $UserCredential = Get-Credential
2. Then issue this command to connect:  $Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection
3. Import the PowerShell commands from the Exchange Server by issuing:  Import-PSSession $Session
4. Now test your connection by issuing: Get-Mailbox and making sure you get output.
4. https://www.cogmotive.com/blog/office-365-tips/create-shared-mailboxes-with-same-alias-at-different-domains-in-office-365 gives us the following steps
1. :  New-Mailbox -Name "Test Shared Mailbox 2" -Alias test_shared2 -Shared -PrimarySMTPAddress test_shared@cogmotivereports.com       Be certain to replace "Test Shared Mailbox 2" with the desired display name, the alias "test_shared2" with the desired alias (prefix before the @ sign) and fix the primary SMTP address with the desired address.
2. Next correct the login name for this mailbox - set-mailbox test_shared2 -MicrosoftOnlineServicesID test_shared@cogmotivereports.com - you may receive the following error WARNING: UserPrincipalName "test_shared1@cogmotivereports" should be same as WindowsLiveID "test_shared@cogmotive.onmicrosoft.com", UserPrincipalName should remain as"test_shared1@cogmotivereports". Which may be safely ignored.
5. Check you work by issuing Get-Mailbox verifying that the new mailbox entry appears.
6. Close your session with Remove-PSSession $Session

This will create a new shared mailbox with the desired alias.  To make certain users send from the desired address and not the placeholder alias (test_shared2) sign into Exchange and go to the shared mailbox.  Edit the email addresses and set the desired alias as the primary address.