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

It is well known that every regular Cayley map $M=\text{CM}(G,X,p)$ 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 $\phi$ on $G$ such that the restriction of $\phi$ to $X$ is $p$. The skew-type of the map $M$ is defined as the index $[G:\text{Ker}\phi ]$, which equals the number of distinct values in ${\mathbb{Z}}_{\mathrm{\mid }\phi \mathrm{\mid }}$ taken by the associated power function $\pi$ of the skew morphism $\phi$. 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.