Using Automator to Move Mail Messages

as you've found out there are no automator actions for this. you can do it with apple script though. you can run that apple script separately or embed it into an automator workflow using "run applescript" action. see this page for the basics on scripting Mail
http://www.mactech.com/articles/mactech/Vol.21/21.09/ScriptingMail/index.html

Why not use the Archive Mailbox selection from Mail's Message menu? Select your Sent mailbox, then use the Archive Mailbox option.

Why do you want to put Sent mail into an archive? The Sent mailbox is the normal place to keep your sent mail.

Mail's rules only work automatically on incoming messages, so there's really no obvious way to do what you want automatically. You could write a rule to move the messages, but you would have to run the rule manually since Mail doesn't have any other way for rules to operate automatically than on incoming messages. Otherwise you would need to accomplish the task using AppleScript. An AppleScript could be tied to a schedule using the OS X cron facility or a launch agent.

You could create a smart mailbox that contains all the items in your inbox AND all the items in your Sent mailbox

You could process that as your smart inbox rather than your actual Mail inbox.

or

You could setup mail in preferences - composing to BCC yourself on every message. This will cause a duplicate of the message to appear in your Inbox

Thank you for this: It was a straight forward and easy solution for me.