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 $\phi$ on $A$ that fixes the identity element of $A$ and for which there exists an integer-valued function $\pi :A\to {\mathbb{Z}}_{\mathrm{\mid }\phi \mathrm{\mid }}$ such that $\phi (xy)=\phi (x){\phi }^{\pi (x)}(y)$ for all $x,y\in A$. The kernel of $\phi$ is the subgroup $\text{Ker}\phi =\{x\in A\mid \pi (x)=1\}$, and the index $[A:\text{Ker}\phi ]$ is called the skew-type of $\phi$. 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.