Finite Groups of Derangements on the n-Cube II

Given k ∈ N and a finite group G, it is shown that G is isomorphic to a subgroup of the group of symmetries of some n-cube in such a way that G acts freely on the set of k-faces, if and only if, gcd(k, |G|) = 2 s for some non-negative integer s. The proof of this result is existential but does give some ideas on what n could be.