Is there an example of a finitely generated subgroup $\Gamma \subset \mathrm{GL}\_n(\mathbb{C})$ such that the group of outer automorphisms $\mathrm{Out}(\Gamma)$ contains finite subgroups of unbounded orders?



Outer automorphisms of finitely generated linear groups
Yes. Furthermore, every recursively presented countable group embeds in such a group. Indeed, first Higman's embedding reduces to proving that every finitely presented group embeds into such a group (if you want all finite groups, it is enough to use a single known f.p. group with all finite groups in it, e.g., Thompson's group $V$).


Now the Rips construction says that for every f.p. group $Q$, there exists a $C'(1/6)$ f.p. small cancelation group $G$ and a finitely generated and normal subgroup $N$ such that $G/N\simeq Q$. Hence $Q$ embeds into $\mathrm{Out}(N)$.


Finally, we use Haglund-Wise-Agol's results, which ensure that every $C'(1/6)$ f.p. small cancelation group is linear over $\mathbf{Z}$.


*Edit:* as [suggested by HJWR](https://mathoverflow.net/questions/447835/outer-automorphisms-of-finitely-generated-linear-groups/447836?noredirect=1#comment1157243_447836) in a comment, it is more direct to directly use the Haglund-Wise "Rips construction" ([GAFA 2008](https://doi.org/10.1007/s00039-007-0629-4)-[other link](http://math.hunter.cuny.edu/olgak/haglund_Wise.pdf)), which directly provides a group that is linear over $\mathbf{Z}$, without need of the Wise machinery plus Agol.


