Pentavalent symmetric graphs admitting transitive non-abelian characteristically simple groups

Let $\mathrm{\Gamma }$ be a graph and let $G$ be a group of automorphisms of $\mathrm{\Gamma }$. The graph $\mathrm{\Gamma }$ is called $G$-normal if $G$ is normal in the automorphism group of $\mathrm{\Gamma }$. Let $T$ be a finite non-abelian simple group and let $G = T^l$with$l\geq 1$. In this paper we prove that if every connected pentavalent symmetric $T$-vertex-transitive graph is $T$-normal, then every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal. This result, among others, implies that every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal except $T$ is one of $57$ simple groups. Furthermore, every connected pentavalent symmetric $G$-regular graph is $G$-normal except $T$ is one of $20$ simple groups, and every connected pentavalent $G$-symmetric graph is $G$-normal except $T$ is one of $17$ simple groups.