By default, emails sent from Shared Mailbox are only saved in the user’s Sent Items folder but ideally, it can be also copied in the Shared Mailbox’s Sent Items folder as well.
To enable this Sent Items Copy feature,
First, connect to Office 365 using:
$userCredential = Get-Credential $Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection Import-PSSession $session
Set-Mailbox emailaddress -MessageCopyForSentAsEnabled $true
Fairly easy, and straightforward.