I know next to nothing about proxying, so we need to wait for someone who speaks Apache (
looking at you, phranque).
how to authorize the proxy at the google site end
Is it your site? That is, obviously it belongs to google, but do you have access to the server config file? I'm guessing no, and this isn't the kind of thing you can do in htaccess. afaik, mod_proxy directives are limited to config or, at most, virtual host.
I assume there's some reason you can't just park the content on your own site and make the problem disappear that way. Can you get an iframe to work if you follow not2easy's suggestion of giving both pages the same protocol?
Incidentally, I tried the [P] flag on my test site, proxying to another domain on the same server. If you don't have use of* mod_proxy, the error you get is:
File does not exist: proxy:http://www.example.com/dir/filename.html
* That is, it may or may not exist, but I can't do anything about it.