Exchange Online: Save Sent Items in Shared Mailbox
Sent items are default not saved to the sent items folder of the shared mailbox, with following commands you can save a copy to the sent items folder of the shared mailbox.
First, login to exchange online with powershell (https://www.productivv.be/2022/12/01/powershell-exchange-online-commands/)
Set-Mailbox <shared mailbox> -MessageCopyForSentAsEnabled $true