Characteristically metacyclic not implies metacyclic derived series

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., characteristically metacyclic group) need not satisfy the second group property (i.e., group with metacyclic derived series)
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Statement

It is possible to have a group $G$ with a Cyclic characteristic subgroup (?) $N$ such that $G/N$ is also cyclic, but such that the abelianization $G/[G,G]$ is not cyclic.

Proof

Example of the dihedral group

Further information: dihedral group:D8

Consider the dihedral group of order eight:

$G:=⟨a,x\mid a^4=x^2=e,xa{x}^{-1}=a^{-1}⟩$.

Let $K$ be the four-element cyclic subgroup $⟨a⟩$. 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.