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 $φ$ on $A$ that fixes the identity element of $A$ and for which there exists an integer-valued function $π:A\to\mathbb{Z}_{|φ|}$ such that $φ(xy)=φ(x)φ^{π(x)}(y)$ for all $x,y\in A$. The kernel of $φ$ is the subgroup $\Kerφ=\{x\in A\mid π(x)=1\}