On Hamilton paths in vertex-transitive graphs of order $10p$

It was shown by Kutnar, Maru\v si\v c and Zhang in 2012 that every connected vertex-transitive graph of order $10p$, where $p$ is a prime and $p\mathrm{\ne }7$, contains a Hamilton path, except for graphs $X$ arising from the action of PSL$(2,s^m)$ on cosets of ${\mathbb{Z}}_s^m⋊{\mathbb{Z}}_{\frac{s^m-1}{10}}$, where $s$ is a prime. In this paper, Hamilton cycles of these exceptions $X$ will be found.