A generalization of the Chermak--Delgado measure on subgroups and its associated lattice

We generalize the Chermak–Delgado measure of a subgroup of a finite group G, µ(H) = |H||CG(H)|, and its associated lattice of subgroups with maximal measure. We consider mappings M of the lattice of all subgroups Sub(G) into itself and define a measure associated to M by setting µ(H) = |H||M(H)|. We investigate under what conditions on M the subgroups with maximal measure form a sublattice of Sub(G). In particular, our focus is on the case where M(H) is a centralizer-like subgroup.