Abelian hereditarily normal implies finite-pi-potentially verbal in finite

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Statement

Suppose G is a finite group and H is an Abelian hereditarily normal subgroup (?) of G. In other words, H is an abelian normal subgroup of G such that the induced action of the quotient group G/H on H is by power automorphisms.

Then, H is a Finite-pi-potentially verbal subgroup (?) of G. In other words, there exists a finite group K containing G such that all prime factors of the order of K also divide the order of G, and such that H is a verbal subgroup of K.

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