F5 - insert http header on redirect and carry it over to the redirected site?

Hi,

  I am trying to achieve the following: F5 pretending to be website1.client.com redirects to website2.client.com and inserting http header on redirect. We need to carry this header over in browser request to website2.client.com

I am able to insert HTTP header using HTTP:redirect:

when HTTP_REQUEST {
  if { [HTTP::host] equals "webserver1.client.com" } 
       {
            HTTP::respond 302 Location "https://webserver2.client.com" thisisheadername thisisheadervalue
          
       } 
}

  I do see my cookie among response headers, however, when going to webserevr2.clinet.com, my cookie not among request headers.

  Is it possible to do? 

Thank in advance.