When you search people/groups at manage-permissions page share presentation webscript is used <blockquote>share/service/components/people-finder/authority-query?authorityType=all&maxResults=100&filter=adm</blockquote>. This webscript uses /api/groups and /api/people repository webscripts.
See <blockquote>/share/WEB-INF/classes/alfresco/site-webscripts/org/alfresco/components/people-finder/authority-query.get.js</blockquote> for details.
You can easily find ajax webscript requests using browser debugger. In Chrome press F12 and go to Network -> XHR tab.