Recognizing $\mathrm{PSL}(2,p)$ in the non-Frattini chief factors of finite groups

Given a finite group $G$, let $P_G(s)$ be the probability that $s$ randomly chosen elements generate $G$, and let $H$ be a finite group with $P_G(s)=P_H(s)$. We show that if the nonabelian composition factors of $G$ and $H$ are $\mathrm{PSL}(2,p)$ for some non-Mersense prime $p\geq 5$, then $G$ and $H$ have the same non-Frattini chief factors.