Email sent from a shared mailbox are not saved to the Sent Items

 Issue :

When a user sends an Email from his delegated (shared) mailbox the Email which sent from the delegated mailbox are stored in user's (who sent mail) sent item instead of the shared mailbox's sent item.

Cause :

When user send a new message from the shared mailbox, Outlook automatically sends from the sender's account. Therefore, messages are stored in the Sent Items folder of the sender's mailbox.

Solution :

Note : Outlook must be configured to run in cached mode for this option to work correctly. This will not work in NEW OUTLOOK mode which does not support cached mode.

1. Click Start, click Run, type regedit, and then click OK.

2. Locate and then click the following registry subkey:

   HKEY_CURRENT_USER\Software\Microsoft\Office\x.0\Outlook\Preferences

    Note :

   The x.0 placeholder represents your version of Office (16.0 = Office 2016, Office 2019, or Office LTSC 2021, 15.0 = Office 2013, 14.0 = Office 2010).

3. On the Edit menu, point to New, and then click DWORD Value.

4. Type DelegateSentItemsStyle, and then press Enter.

5. Right-click DelegateSentItemsStyle, and then click Modify.

6. In the Value data box, type 1, and then click OK.

7. Exit Registry Editor.

8. Close and open the outlook and try to send EMail again.

Source : https://learn.microsoft.com/en-us/exchange/troubleshoot/user-and-shared-mailboxes/sent-mail-is-not-saved

https://learn.microsoft.com/en-us/outlook/troubleshoot/email-management/email-remains-in-the-outbox-when-you-use-the-deleg