Classification of regular Cayley maps of skew-type three on semidihedral groups

It is well known that every regular Cayley map $M = \CM(G,X,p)$ on a finite group $G$ with respect to an inverse-closed generating set $X$ of $G$ and a specified cyclic permutation $p$ on $X$ corresponds to a skew morphism $\varphi$ on $G$ such that the restriction of $\varphi$ to $X$ is $p$.

The skew-type of the map $M$ is defined as the index $[G:\Ker \varphi]$, which equals the number of distinct values in $\mathbb{Z}_{|\varphi|}$ taken by the associated power function $\pi$ of the skew morphism $\varphi$.

In this paper, we develop a covering theory of skew morphisms and as an application we provide a classification of regular Cayley maps of skew-type three on the semidihedral groups.