I have been trying what is missing on the production server and I just cannot find anything about it.
When a Web Service request is received, the logic goes in this SoapExtension class. However, the inherited methods are not being used. So, something is missing on the production server as far as .NET modules which is causing this class to not recognize its inherited method.
So, again, this is working really well on the development server. And, so far, everything appears the same on all servers. I have checked the .NET Framework version, the IIS version, the Windows updates, the IIS configuration and many other things like that.