This has nothing to do with Transfer. For this you would use
HttpHandlers (or whatever it is named).
Actually, you don't needs those at all, you only need to use the
UrlRewrite method. For some reason I've never worked out, this extremely
simple and very useful technique is made horribly complex in almost
every article and sample going by using handlers.
As long as the fake URL ends in .aspx, it will be handed to ASP.NET for
processing. You just add code to your global.asax to check for the fake
URL and use UrlRewrite to call the real page. The browser still shows
the fake URL.
HTH