If you arrange the edge skeletons of a convex regular pentagon and a regular pentagram in the right way and connect each vertex of the former to that of the latter lying under it, the result (the Petersen graph) is significantly more symmetric than you might expect, and in fact is a symmetric graph. Does the same thing happen if you use a convex regular icosahedron (or great dodecahedron) and a small stellated dodecahedron (or great icosahedron) instead of the pentagons?

If so, the full automorphism group should be of order 2880.

If you arrange the edge skeletons of a convex regular pentagon and a regular pentagram in the right way and connect each vertex of the former to that of the latter lying under it, the result (the Petersen graph) is significantly more symmetric than you might expect, and in fact is a symmetric graph. Does the same thing happen if you use a convex regular icosahedron (or great dodecahedron) and a small stellated dodecahedron (or great icosahedron) instead of the pentagons?

If so, the full automorphism group should be of order 2880 (Edit: or 1440).