Ordinary $G$-permutable subgroups of finite simple groups

We completely solve part (a) of Problem 17.112 in the Kourovka Notebook. We prove that a non-abelian finite simple group $G$ has a proper non-trivial ordinary $G$-permutable subgroup if and only if $G$ is isomorphic to $\mathrm{P}\mathrm{S}{\mathrm{L}}_2(\mathrm{q})$ where $q\ge 4$ is a prime power and $q\equiv ̸3(\mathrm{m}\mathrm{o}\mathrm{d}4)$, ${}^2{B}_2(2^{2a+1})$ where $a\ge 1$, $J_1$, ${\mathrm{S}\mathrm{p}}_4(4)$, or ${\mathrm{P}\mathrm{S}\mathrm{p}}_4(p)$ where $p$ is prime and $p\equiv 1,13,17(\mathrm{m}\mathrm{o}\mathrm{d}24)$. Moreover, we classify all proper non-trivial ordinary $G$-permutable subgroups of these groups. Every other non-abelian finite simple group $G$ has no proper non-trivial ordinary $G$-permutable subgroup.