Add Google Programmable Search Engine - Google Sites
Display Google's Programmable Search Engine (formerly Custom Search Engine) on a public new Google Site
Add CSE to new public Google Site
Go to https://cse.google.com/cse/ and choose the search engine you wish to embed
Use the Get code button to get the code for the search engine you want to embed
Edit your new Google Site embed the custom code
Google CSE can only index public Google Sites, sites published to your domain only cannot be access by Google CSE.