TITLE: A group which is $\mathbb{Z}$-by-finite but not finite-by-$\mathbb{Z}$
QUESTION [6 upvotes]: I found a lemma which states: 

If a group $G$ is finite-by-$\mathbb{Z}$, then $G$ is
  $\mathbb{Z}$-by-finite.

I was wondering if the converse is true, i.e. is it true that

if a group G is $\mathbb{Z}$-by-finite, it's also finite-by-$\mathbb{Z}$?

I suspect not, but as I'm only starting to begin to understand the theory, I can't come up with a counterexample. 
Thanks in advance!

EDIT: A brief explanation of $\mathcal{P}$-by-$\mathcal{Q}$:  Let $\mathcal{P}$ and $\mathcal{Q}$ be 2 properties of groups. Then a group $G$ is $\mathcal{P}$-by-$\mathcal{Q}$ if there exist a normal subgroup $N$ of $G$ such that $N$ has property $\mathcal{P}$ and $G/N$ has property $\mathcal{Q}$.
The line "has property $\mathbb{Z}$" means: being isomorphic to the infinite cyclic group.

REPLY [2 votes]: The dihedral group is in a sense the only exception: a group is $\mathbf{Z}$-by-finite if and only if it is finite-by-$\mathbf{Z}$ or finite-by-$D_\infty$.