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](https://en.wikipedia.org/wiki/Small_cancellation_theory) 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](https://berstein.wordpress.com/2011/02/27/the-rips-construction-i) and [here](https://berstein.wordpress.com/2011/03/02/the-rips-construction-ii). 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 $Out(N)$ contains $Q$ and thus has infinite vcd. 

I am not sure how to find examples where $Aut$ has infinite vcd. 

[1] <cite authors="Rips, E.">_Rips, E._, [**Subgroups of small cancellation groups**](https://doi.org/10.1112/blms/14.1.45), Bull. Lond. Math. Soc. 14, 45-47 (1982). [ZBL0481.20020](https://zbmath.org/?q=an:0481.20020).</cite>

**Edit.** A modification of the Rips construction in [2] yields examples where $Out(N)\cong Q$ even though it is not need for your question. 

[2] <cite authors="Bumagin, Inna; Wise, Daniel T.">_Bumagin, Inna; Wise, Daniel T._, [**Every group is an outer automorphism group of a finitely generated group.**](https://doi.org/10.1016/j.jpaa.2004.12.033), J. Pure Appl. Algebra 200, No. 1-2, 137-147 (2005). [ZBL1082.20021](https://zbmath.org/?q=an:1082.20021).</cite>