Send fee type to Authorize.net along payment
By fee type I am assuming you mean financial type, but if I am not mistaken, financial type information is not transferred to any payment processor - that all remains within CiviCRM, so to get that information from a report you generate from Authorize.net would not be possible without having something developed specifically for that purpose.

If I have misunderstood and you are referring to a report that you run from within CiviCRM, it could be that you either have a custom report or renamed a built-in report - if it is the latter, please let me know the name of the report that you are using.

Hope this helps!
Tamar