This can be done using acf_form() http://www.advancedcustomfields.com/resources/acf_form/
Create the options page and the field groups you want on it and then for the front end form use options as the post_id value and list the field groups that you want to show.
$args = array(
'post_id' => 'options',
'field_groups' => array(3199) // this is the ID of the field group
);
acf_form($args);
Follow the rest of the instructions for using acf_form()
