You have to type in the name of the group like you're tagging them. Once you begin typing the name it should come up on your screen and allow you to choose that group. Some FB groups do not allow shares, so you'll have to read the group rules and find out.
I had this problem and finally figured it out.
On the share link select 'write post'
Up the top, in the middle it should say 'your timeline'
Select this and a dropdown list appears one of which is 'Groups'
Select this and you should be able to share to a group.