Sorry, it is Rips' and not Mikhailova's construction: I should not comment when I am half-asleep.
Let me start with the Rips construction.
Let $Q$ be a finitely presented group. Rips in [1] constructed $C'(1/\lambda)$-small cancellation groups $G$ (with arbibtarily large $\lambda$) and normal finitely generated subgroups $N< G$ such that $G/N\cong Q$. For $\lambda\ge 7$ the group $G$ will be hyperbolic.
A nice exposition of the Rips construction and its generalizations can be found in these two blog-posts: here and here. Actually, the Rips construction is quite flexible and one can make choices so that no defining relator of $G$ is a proper power; hence, the presentation complex of $G$ is aspherical. In particular, $G$ is torsion-free and is 2-dimensional. The subgroup $N$, therefore, is also 2-dimensional. However, the group $Q$ can be taken to have infinite virtual cohomological dimension.
We, thus, obtain a finitely generated 2-dimensional group $N$ such that $Out(N)\cong Q$ has infinite vcd.
I am not sure how to find examples where $Aut$ has infinite vcd.
[1] Rips, E., Subgroups of small cancellation groups, Bull. Lond. Math. Soc. 14, 45-47 (1982). ZBL0481.20020.