This is one of the way to add google site search to websites:
<form action="https://www.google.com/search" class="searchform" method="get" name="searchform" target="_blank">_x000D_
<input name="sitesearch" type="hidden" value="example.com">_x000D_
<input autocomplete="on" class="form-control search" name="q" placeholder="Search in example.com" required="required" type="text">_x000D_
<button class="button" type="submit">Search</button>_x000D_
</form>
_x000D_