Classify skew morphisms of arbitrary power-of-two skew-type on cyclic 2-groups

Extend the methods used to classify skew morphisms of skew-type 4 on the cyclic additive groups \(\mathbb{Z}_{2^{e+2}}\) to classify skew morphisms of skew-type \(2^a\) on \(\mathbb{Z}_{2^{e+2}}\) for arbitrary \(a\), and derive a closed-form enumeration of these skew morphisms.

Background

The paper completes the classification and enumeration of skew morphisms of skew-type 4 on cyclic 2-groups, building on earlier classifications for skew-types at most 3. The authors identify the extension to arbitrary skew-types $2^a$ as a natural unresolved generalization of their covering and congruence-based methods.

The open problem has two linked components: obtaining a classification for every power-of-two skew-type $2^a$, and producing a closed-form formula for the number of such skew morphisms.