Facebook added the ability to set the invitation to your groups automatically if your Facebook page is admin of the group.
All you need to do is to navigate to your Facebook fan page, then open the groups tab, then click on manage invites, then switch on the automatic invitation for a group of yours.
You can auto-invite people who like your page to one of your groups. Here is a video to see the steps.
I hope you benefit from this information told in brief.