Iterated agemo subgroup of p-group

Definition
Suppose $$p$$ is a prime number and $$G$$ is a p-group. A subgroup $$H$$ of $$G$$ is an iterated agemo subgroup of $$G$$ if there exists a sequence $$a_1,a_2,\dots,a_r$$ such that:

$$H := \mho^{a_1}(\mho^{a_2}(\mho^{a_3}(\dots(\mho^{a_r}(G)))\dots)$$.