This is happening because you are replying as yourself, not as that mailbox. To have the reply go to the sent items of the group mailbox, insert the name of the group mailbox in the From field of the e-mail when replying.
If you don't currently have the From field showing, simply click on View > From Field from within an Outlook e-mail.
HTH
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Mike
Why make it simple and efficient when it can be complex and wonderful?