When you sync to Outlook (2016 and newer) using MAPI/EWS the out of office autoresponder should sync between SmarterMail and Outlook. When you put a message in the "Inside My Organization" tab in Outlook, that should sync to the "Response to Send to Domain Users" section in webmail. And the same thing should happen between the Outlook tab "Outside My Organization" and Response to Send to Everyone Else section in webmail.
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For faster and more reliable delivery, add email@example.com to your trusted senders list in your email software.