  A graph \Gamma is said to be {\em symmetric} if its automorphism group
\Aut(\Gamma) is transitive on the arc set of \Gamma. Let $G$ be a finite
non-abelian simple group and let \Gamma be a connected pentavalent symmetric
graph such that G\leq \Aut(\Gamma). In this paper, we show that if $G$ is
transitive on the vertex set of \Gamma, then either G\unlhd \Aut(\Gamma) or
\Aut(\Gamma) contains a non-abelian simple normal subgroup $T$ such that $G\leq
T$ and $(G,T)$ is one of $58$ possible pairs of non-abelian simple groups. In
particular, if $G$ is arc-transitive, then $(G,T)$ is one of $17$ possible
pairs, and if $G$ is regular on the vertex set of \Gamma, then $(G,T)$ is one
of $13$ possible pairs, which improves the result on pentavalent symmetric
Cayley graph given by Fang, Ma and Wang in 2011.
