Combine internal search with Google custom search?

The standard search results can also be formatted with a theme file, by default search.php. You don’t need to create a “fake” page for that.

If you don’t include any content from the page, and only use the theme file, you can just rename searchpage.php to search.php and it will work. If you do have some content that comes from a page written in the admin area, then you will have to query for this yourself (using get_posts() or another way), because the standard loop will only contain the search results, not that specific page.

So if you want to add the internal WordPress search results to the page, just add a classic loop at the top, and this should display the results.