How can only FB Closed Group members log into the members section of a website?

Please can anyone help.

I am getting desperate as searching WordPress Forums and the Internet has proved absolutely fruitless.

As one of the admins for a school alumni Facebook closed group of 600+ members, I am in the process of setting up a website to work along side its FB Closed Group.
To save members having to register again to gain access to the members section of the website it would make life a lot simpler if they could just simply click the ‘Login with Facebook’ button but that’s where the problem lies:
How can only FB Closed Group members log into the members section of a website while blocking everyone one else?
There are plenty of ‘Login with Facebook’ plugins available but these allow for Joe Public (anyone) to login.
I have searched high and low for a plugin, php script or FB Graph API settings that will allow only FB Closed Group members to login while sending none members to another page but to no avail.
2 well established Facebook login plugin authors have said that since Facebook has revamped its API (killed its API, as one put it) it’s something that they plan to look at towards the end of the year or early next.

Can anyone please offer any help/suggestions on how to allow only Facebook closed group members to login to the members section of a website.

Thank you in advance for any help