We discuss a way to construct permutation actions on the equivariant cohomology of various kinds of subvarieties of the flag variety. To do this, we review the Goresky-Kottwitz-MacPherson (GKM) approach to equivariant cohomology, which describes the equivariant cohomology of a suitable variety in terms of the variety's moment graph (a combinatorial graph obtained from the moment map). The permutation action on the equivariant cohomology can be described directly in terms of (combinatorial) graph automorphisms. Some examples of the kinds of subvarieties whose equivariant cohomology carries these permutation representations include some Schubert varieties and some Hessenberg varieties. We give examples of some of the representations that can be constructed, as well as many open questions.