Automorphisms of the cube nd

Consider a hypergraph $H_n^d$ where the vertices are points of $d$-dimensional combinatorial cube $n^d$ and edges all sets $n$ such that they in one line. We study structure group automorphisms $H_n^d$, i.e., permutations preserving edges. In this paper we provide complete characterization. Moreover, consider Colored Cube Isomorphism problem deciding whether for two colorings there exists an automorphism colors. show is ${\sf GI}$-complete.