$2$-groups behaving as automorphism groups of regular $3$-polytopes

In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schlafli 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].