A characterization of the simple group $PSL_5(5)$ by the set of its element orders

Let $G$ be a finite group and let $\omega(G)$ denote the set of the element orders of $G$. For the simple group $PSL_5(5)$ we prove that if $G$ is a finite group with $\omega(G)=\omega(PSL_5(5))$, then either $G\cong PSL_5(5)$ or $G\cong PSL_5(5):\langle\theta\rangle$ where $\theta$ is a graph automorphism of $PSL_5(5)$ of order 2.