You must set up a HTTP reverse proxy server that does the authentication for you. No need to hack any software.
Your reverse proxy listens on some socket (e.g. proxy:8080) and forwards requests to your actual application A, inserting the headers.
client_B ----> http://proxy:8080 -----> http://server_A:80
Nginx is lightweight, high performance and easy to set up. And it's easy to find docs online for what you want. See for example http://wiki.apache.org/couchdb/Nginx_As_a_Reverse_Proxy