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\PreferencesNote 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.

https://support.microsoft.com/en-us/help/2843677/messages-sent-from-a-shared-mailbox-aren-t-saved-to-the-sent-items-fol