Iterated agemo subgroup not implies agemo subgroup

Statement
It is possible to have a group of prime power order $$G$$ and an iterated agemo subgroup $$H$$ of $$G$$ such that $$H$$ is not an agemo subgroup of $$G$$.

Example for $$p = 2$$
Let $$G$$ be the group that we can construct as SmallGroup(64,34) using GAP. Then, we have $$\mho^1(G)$$ is a group of order $$16$$ isomorphic to the particular example::direct product of D8 and Z2, and $$\mho^2(G)$$ is trivial because $$G$$ has exponent $$4$$.

The group $$\mho^1(\mho^1(G))$$ is a cyclic subgroup of order two, which is thus an iterated agemo subgroup not equal to any of the agemo subgroups.

The same observations hold for SmallGroup(64,35).

GAP implementation of proof
gap> G := SmallGroup(64,34);  gap> IdGroup(Agemo(G,2,1)); [ 16, 11 ] gap> G := SmallGroup(64,34);  gap> H := Agemo(G,2,1); Group([ f3, f4, f5, f6 ]) gap> K := Agemo(H,2,1); Group([ f6 ]) gap> L := Agemo(G,2,2); Group([ of ... ]) gap> IsTrivial(L); true gap> IdGroup(K); [ 2, 1 ] gap> IdGroup(H); [ 16, 11 ]