How to add an extra form action on the user login page?
Use hook_form_FORM_ID_alter() in order to modify the specific form.
You can add the action or the link there.
The link can be added using the #prefix,#suffix attributes of either the form or of a specific element.

Prefix example:

$form['#prefix'] = l(t('Link text'), 'my link');


Action example:

$form['actions']['add'] = array(
  '#type' =&gt; 'submit',
  '#value' =&gt; t('Add'),
  '#submit' =&gt; array('_my_callback_function'),
);