If you experience no effects such as the missed login after clicking on buttons for the social authentication, the problem could be related to the Instant Facebook Login module assignment.
In more details, it could be the case that when you click on the social buttons the authentication starts but after that you are redirected back to your website the Joomla login doesn't take place at all.

In such case ensure the following:

• The module Instant Facebook Login must be assigned directly to a valid template position. This means that you can't embed the module inside contents using {loadmodule} or {loadposition}. if you embed the module in such way the social login won't work.

• If you are using Joomla 4 and later, ensure that the plugin 'System - HTTP Headers' is unpublished or properly configured in order to allow Facebook to store cookies. Be careful that this plugin, even with default settings, could fully prevent the login via the Facebook SDK.

• The module Instant Facebook Login must be assigned to the page used for the redirection after the social authentication. If you redirect to a page where the module is not assigned the social login won't work.

• In order to allow the social login to work correctly the module MUST be published and assigned to the redirection page of your choice after the login with Facebook or Twitter. If the social login module is not published on the redirection page the login phase won't be processed and it will fail.
  Moreover the Google login requires that the module is published on the site home page too. Finally the LinkedIn login requires that all pages of your website where the login can take place will be registered in the LinkedIn app.

• The plugin 'Authentication - Joomla' must stay published as by default for every Joomla installation.