Show that .
Recall that , and let be an automorphism of .
We have . Then . There are 6 elements of order 4 in , so we have 6 choices for . Note that if we choose , then is not surjective, hence not an isomorphism. Two of the six elements of order 4 are in , so we have 4 choices for . Now is determined by its image at and , so that .