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 $Out(N)$ contains $Q$ and thus 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.

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 $Out(N)$ contains $Q$ and thus 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.

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] Bumagin, Inna; Wise, Daniel T., Every group is an outer automorphism group of a finitely generated group., J. Pure Appl. Algebra 200, No. 1-2, 137-147 (2005). ZBL1082.20021.

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.

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 $Out(N)$ contains $Q$ and thus 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.