Abelian hereditarily normal implies finite-pi-potentially verbal in finite
Suppose is a finite group and is an Abelian hereditarily normal subgroup (?) of . In other words, is an abelian normal subgroup of such that the induced action of the quotient group on is by power automorphisms.
Then, is a Finite-pi-potentially verbal subgroup (?) of . In other words, there exists a finite group containing such that all prime factors of the order of also divide the order of , and such that is a verbal subgroup of .