Strong Gelfand pairs of the sporadic groups and their extensions

A strong Gelfand pair $(G,H)$ is a finite group $G$ and a subgroup $H$ where every irreducible character of $H$ induces to a multiplicity-free character of $G$. We determine the strong Gelfand pairs of the sporadic groups, their automorphism groups, and their covering groups. We also find the (strong) Gelfand pairs of the generalized Mathieu groups, the Tits group, and the automorphism group of the Leech Lattice.