Earlier quoted context omitted.
The problem is that the search query does not go in the URL in the Lite version. That really sucks. It makes it useless in the browser history as well. In fact, because all the URLs are the same on each query, they're not even added as separate entries in the history.
They don't? Search term(s) are the URL parameter: ?q= Which does show in my history AFAICT.
In the search form, it uses POST instead of GET to perform the searches.