On the existence of $G$-permutable subgroups in simple sporadic groups

A subgroup $A$ of a group $G$ is $G$-permutable in $G$ if for every subgroup $B\leq G$ there is $x\in G$ satisfying $AB^x=B^xA$. 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 contains Problem 17.112: Which finite nonabelian simple groups $G$ possess a proper (hereditarily) $G$-permutable subgroup? We answer the question for simple sporadic groups.