Smooth skew morphisms on semi-dihedral groups

A permutation \(\varphi \) $\phi$ on a group G is called a skew morphism of G if \(\varphi (1) = 1\) $\phi (1)=1$ , and there exists an integer-valued function \(\pi : G \rightarrow Z_m\) $\pi :G\to Z_m$ , where m is the order of \(\varphi \) $\phi$ , such that \(\varphi (ab) = \varphi (a)\varphi ^{\pi (a)}(b)\) $\phi (ab)=\phi (a){\phi }^{\pi (a)}(b)$ , for all \(a, b\in G\) $a,b\in G$ . A skew morphism \(\varphi \) $\phi$ is smooth if the associated power function \(\pi \) $\pi$ of \(\varphi \) $\phi$ takes constant values on each orbit of \(\varphi \) $\phi$ . In this paper, we shall classify the smooth skew morphisms of semi-dihedral groups.