Of all the 1-skeleton graphs of finite regular polytopes, exactly one is more symmetric than the polytope or polytopes it is derived from: that of the small stellated 120-cell and great grand 120-cell. In particular, the polytopes have as their symmetry group the $H_4$ Coxeter group, isomorphic to $2\cdot(A_5\times A_5):2$, while the skeleton has an index-2 overgroup of that. What is this group?
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