  A graph is said to be symmetric if its automorphism group is transitive on
its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al.
(Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric
graphs of order $4pq$. In this paper, we shall generalize this result by
determining all connected pentavalent symmetric graphs of order four times an
odd square-free integer. It is shown in this paper that, for each of such
graphs $\it\Gamma$, either the full automorphism group ${\sf Aut}\it\Gamma$ is
isomorphic to ${\sf PSL}(2,p)$, ${\sf PGL}(2,p)$, ${\sf
PSL}(2,p){\times}\mathbb{Z}_2$ or ${\sf PGL}(2,p){\times}\mathbb{Z}_2$, or
$\it\Gamma$ is isomorphic to one of 8 graphs.
