On Page Google Search Results
One of the biggest pet peeves of Google’s search bar that you can put on your pages to search Google’s index or your own site is that it brings you to an external Google page, rather than keeping people on your own sites.
Google has rectified this problem recently in making it so that the results show up directly on your own site. Some people have reported problems trying to get this work, especially with WordPress. I am not sure of the exact problem, but no doubt there will be tweaks, fixes, and hacks to get this improved revenue producing tool working on every type of site.