  A skew-morphism of a finite group $G$ is a permutation $\s$ on $G$ fixing the
identity element, and for which there exists an integer function $\pi$ on $G$
such that $\s(xy)=\s(x)\s^{\pi(x)}(y)$ for all $x,y\in G$. It has been known
that given a skew-morphism $\s $ of $G$, the product of $\lg \s \rg$ with the
left regular representation of $G$ forms a permutation group on $G$, called the
skew-product group of $\s$. The skew-morphism was introduced as an algebraic
tool to investigate regular Cayley maps. In this paper, the skew-product groups
are characterized, for all skew-morphisms of finite nonabelian
characteristically simple groups (see Theorem 1.1) and correspondingly the
Cayley maps on these groups are characterized (see Theorem 1.5).
