n order to save messages sent from a shared mailbox in the sent items of the shared mailbox, modify the Registry for your Outlook:

1. Click Start, click Run, type Regedit, and then click OK.
2. Navigate to HKEY_CURRENT_USER\Software\Microsoft\Office\x.0\Outlook\Preferences

The x.0 depends on your Outlook version: 16.0 = Outlook 2016, 15.0 = Outlook 2013, 14.0 = Outlook 2010

1. Right-click on Edit menu, navigate to New > DWORD Value
2. Type DelegateSentItemsStyle, press Enter
3. Double-click DelegateSentItemsStyle, type in 1 in the Value Data box, click OK