Answer
Configure the actual Selector element with filter fields: go to the Selectors section under All Elements, open your existing selector (or create a new one), and add the fields you want users able to filter/search by. Once filter fields are added and the page is saved and published, click the magnifying-glass icon on the selector to reveal the filtering options — the search capability comes from the Selector's own filter-field configuration, not a separate setting on the Group content itself.