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).
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In Finitely generated groups with virtually free automorphism groups, 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.