  In this paper, we classify regular polytopes with automorphism groups of
order $2^n$ and Schl\"afli types $\{4, 2^{n-3}\}, \{4, 2^{n-4}\}$ and $\{4,
2^{n-5}\}$ for $n \geq 10$, therefore giving a partial answer to a problem
proposed by Schulte and Weiss in [Problems on polytopes, their groups, and
realizations, Periodica Math. Hungarica 53(2006) 231-255].
