There are only two ways to remove a stuck auger. The first is through the opening it entered by reversing direction, which you probably already tried.
The second is to remove the toilet and see what's holding it in place from the bottom side and either using an auger to push it back or a long screw driver to dislodge the item its stuck on.
Whatever is inside the toilet is holding it in place. Pushing the object back up through the hole using a flashlight and screwdriver, usually gets results. Of course then you have to reset the toilet with a new bowl wax. Make sure you bring in a piece of cardboard to set the toilet on when you turn it upside down and make sure you get all of the water and other stuff out of the toilet.