### Finite Groups of Derangements on the $n$-Cube II

#### Abstract

Given $k\in \mathbb{N}$ 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.