Let $V$ be a virtually cyclic group.
Then is $Aut(V)$ also a virtually cyclic group?


This is true when $V$ is a finite group (zero-ended) and when $V = C\_\infty, D\_\infty$ (both two-ended).



Automorphism groups of virtually cyclic groups
In [Finitely generated groups with virtually free automorphism groups](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=3078536), by M.R. Pettet, it is proved in theorem 3.4 that the automorphism group $Aut(G)$ of a finitely generated group is virtually cyclic if and only if both $Z(G)$ and $G/Z(G)$ are virtually cyclic.


