Verbal not implies iterated agemo

Statement
It is possible to have a group of prime power order $$G$$ and a verbal subgroup (hence, a fact about::verbal subgroup of group of prime power order) $$H$$ of $$G$$ that is not an iterated agemo subgroup of $$G$$.

Proof
Suppose $$p$$ is an odd prime. Consider the group $$G := U(3,p)$$, which is the unique (up to isomorphism) non-abelian group of order $$p^3$$ and exponent $$p$$.

We define $$H := [G,G]$$, so $$H$$ is the commutator subgroup of $$G$$. Clearly, $$H$$ is a verbal subgroup of $$G$$. On the other hand, $$H$$ is not an iterated agemo subgroup of $$G$$, because $$\mho^1(G)$$ is trivial, so the only agemo subgroups (and hence the only iterated agemo subgroups) are the whole group and the trivial subgroup.