Classification and enumeration of skew morphisms of skew-type four on cyclic $2$-groups

A skew morphism on a finite group $A$ is a permutation $\varphi$ on $A$ that fixes the identity element of $A$ and for which there exists an integer-valued function $π:A\to\mathbb{Z}_{|\varphi|}$ such that $\varphi(xy)=\varphi(x)\varphi^{π(x)}(y)$ for all $x,y\in A$. The kernel of $\varphi$ is the subgroup $\Ker\varphi=\{x\in A\mid π(x)=1\}$, and the index $[A:\Ker\varphi]$ is called the skew-type of $\varphi$. In this paper we construct, classify and enumerate the skew morphisms of skew-type four on cyclic $2$-groups. Our main results give explicit formulas for all such skew morphisms and closed-form expressions for their numbers.