Copy-pasted for future reference from Microsoft docs website.

Problem
Assume that you're using Microsoft Outlook 2010 or a later version, and you've been delegated permission to send email messages as another user or on behalf of another user from a shared mailbox. However, when you send a message as another user or on behalf of the user, the sent message isn't saved to the Sent Items folder of the shared mailbox. Instead, it's saved to the Sent Items folder of your mailbox.

Cause
In Office 365, shared mailboxes don't require a license and can't be added to Outlook as an independent mailbox. You can't sign in to a shared mailbox. Instead, you sign in to your own mailbox, and then you open the shared mailbox. When you send or reply to a new message from the shared mailbox, Outlook automatically sends or replies from the sender's account. Therefore, messages are stored in the Sent Items folder of the sender's mailbox.

Method 1: Configure the mailbox to save a copy of the message to the Sent Items folder of the shared mailbox in Exchange Online or on-premises Exchange Server

Exchange Online in Office 365 or Exchange Server 2013 Cumulative Update 9 or later update

Cumulative Update 9 for Exchange Server 2013 introduced a new feature that lets administrators configure the Sent Items folder to which a message is copied. For more information, see Exchange Blog - Want more control over Sent Items when using shared mailboxes?

Using Exchange PowerShell, for emails Sent As the shared mailbox, run the following cmdlet:

set-mailbox <mailbox name> -MessageCopyForSentAsEnabled $True

Using Exchange PowerShell, for emails Sent On Behalf of the shared mailbox, run the following cmdlet:

set-mailbox <mailbox name> -MessageCopyForSendOnBehalfEnabled $True

Exchange Server 2010 Service Pack 2 Update Rollup 4 or later update

Update Rollup 4 for Exchange Server 2010 Service Pack 2 introduced a new Exchange PowerShell cmdlet to configure the Sent Items folder to which a message is copied. Because this new feature is handled by the server running Exchange Server, Outlook can be configured in either online or cached Exchange mode. However, this feature works only if the Outlook DelegateSentItemsStyle registry (Method 2 below) value is disabled.

For more information about the Set-MailboxSentItemsConfiguration cmdlet, see the following Microsoft Knowledge Base article:

2632409 - Messages sent by using the "Send As" and "Send on behalf" permissions are only copied to the Sent Items folder of the sender in an Exchange Server 2010 environment.

Note
The MessageCopyForSentAsEnabled and MessageCopyForSendOnBehalfEnabled settings are not supported if the user mailbox and shared mailbox are located in different environments (cloud and on-premises). The settings are supported only if both mailboxes are in the same environment (cloud or on-premises).

Method 2: Set the DelegateSentItemsStyle registry value on the Outlook client

Note
Outlook must be configured to run in cached mode for this option to work correctly. For more information, see the following Microsoft Knowledge Base article:

2703723 - Email remains in the Outbox when you use the DelegateSentItemsStyle registry value.

If you're running Outlook 2010, install the Outlook 2010 hotfix package dated December 14, 2010. Then, follow these steps:

For more information about this hotfix package, see the following Microsoft Knowledge Base article:

2459115 - Description of the Outlook 2010 hotfix package (outlook-x-none.msp): December 14, 2010

Note
If you're running Outlook 2013 or a later version, you don't need to install any hotfixes.

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