Pentavalent symmetric graphs of order four times an odd square-free integer

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 $\Gamma$, either the full automorphism group $\mathsf{A}\mathsf{u}\mathsf{t}\Gamma$ is isomorphic to $\mathsf{P}\mathsf{S}\mathsf{L}(2,p)$, $\mathsf{P}\mathsf{G}\mathsf{L}(2,p)$, ${\sf PSL}(2,p){\times}$\mathbb{Z}$_2$or${\sf PGL}(2,p){\times}$\mathbb{Z}$_2$,or$\it\Gamma$ is isomorphic to one of 8 graphs.