What protocol is going over the TCP/IP stream?
What is the program at the other end?

Presumably it already expects the authentication somehow? If so why isn't this authentication provided by A?

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Not 100% sure, but I think it's HTTP.

Some kind of server software for an app he is building.

I guess it could be, if computer B does not alter the data in any way, shape or forum, and just passes the information directly on to C.

Perhaps if there is no such program out there, there is some way to hack a broadband router to do this?