On the existence of hereditarily $G$-permutable subgroups in exceptional groups $G$ of Lie type

A subgroup $A$ of a group $G$ is $G$-permutable in $G$ if for every subgroup $B\leq G$ there exists $x\in G$ such that $AB^x=B^xA$. A subgroup $A$ is hereditarily $G$-permutable in $G$ if $A$ is $E$-permutable in every subgroup $E$ of $G$ which includes $A$. The Kourovka Notebook has Problem 17.112(b): Which finite nonabelian simple groups $G$ possess a proper hereditarily $G$-permutable subgroup? We answer this question for the exceptional groups of Lie type. Moreover, for the Suzuki groups $G\cong{^2 \operatorname{B}_2}(q)$ we prove that a proper subgroup of $G$ is $G$-permutable if and only if the order of the subgroup is $2$. In particular, we obtain an infinite series of groups with $G$-permutable subgroups.