Quotient-iterated omega subgroup of group of prime power order

Definition
Suppose $$p$$ is a prime number and $$G$$ is a finite $$p$$-group, so $$G$$ is a group of prime power order. A subgroup $$H$$ of $$G$$ is termed a quotient-iterated omega subgroup of $$G$$ if there exist nonnegative integers $$a_1,a_2,\dots,a_r$$ and normal subgroups $$A_0,A_1,A_2,\dots,A_r$$ of $$G$$, where $$A_0$$ is trivial, $$A_r = H$$, and $$A_i/A_{i-1} = \Omega^{a_i}(G/A_{i-1})$$.