On semigroups whose idempotent-generated subsemigroup is aperiodic

We show that if S is a finite semigroup with aperiodic idempotent-generated subsemigroup and H is a pseudovariety of groups, then the sequence of iterated H-kernels of S stops at the idempotent-generated subsemigroup if and only if each subgroup of S belongs to the wreath product closure of H. Applications are given to Mal'cev products.