Characteristically metacyclic not implies metacyclic derived series

Statement
It is possible to have a group $$G$$ with a fact about::cyclic characteristic subgroup $$N$$ such that $$G/N$$ is also cyclic, but such that the abelianization $$G/[G,G]$$ is not cyclic.

Example of the dihedral group
Consider the dihedral group of order eight:

$$G := \langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle$$.

Let $$K$$ be the four-element cyclic subgroup $$\langle a \rangle$$. Then, both $$K$$ and $$G/K$$ are cyclic, but the abelianization $$G/[G,G]$$ is isomorphic to a Klein four-group, which is not cyclic.