  Let $\Gamma$ be a graph and let $G$ be a group of automorphisms of $\Gamma$.
The graph $\Gamma$ is called $G$-normal if $G$ is normal in the automorphism
group of $\Gamma$. 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.
