Can an HTTPs page be reused for searching?

I have a main page that users visit via https. From that page, they can do a google search. However, the search results is displayed via h开发者_JAVA百科ttp and I no longer have control over the search page. Is there a way to 'trap' the results and send it over a custom https page so the users results can be private?

Update: I guess the real question is can they browse the web through my website and since my site is https, get a private browsing session via my website.

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You can make use of Google AJAX Search API to integrate search results in your own page: http://code.google.com/apis/ajaxsearch/