On minimal faithful permutation representations of finite groups

The minimal faithful permutation degree mu>(*) over bar * (G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group S-n. Let N be a normal subgroup of a finite group G. We prove that mu>(*) over bar * (G/N) less than or equal to mu>(*) over bar * (G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate the structure of a (hypothetical) minimal counterexample to this conjecture.