If you have a Rubik's cube where all the edges flip on each
quarter-turn, you can solve it by using bifocals when it's odd.
I noticed this while drawing a Hasse diagram of the subgroups of M. It
turns out that M has a similar automorphism, where the odd elements are
reflected through the center of the cube.
If anyone wants the Hasse diagram, I can send it--it takes about 30
minutes to draw in the lines, for which directions are included.
If you know whether there are other outer automorphisms of M, please
let me know.