Let $G$ be a countable discrete group, $C_r^*(G)$ its reduced $C^*$-algebra. We say that $G$ has stable rank 1 if $C_r^*(G)$ has stable rank one, that is, the set of invertible elements is dense in $C_r^*(G)$.

Finite groups and non-cyclic torsion free hyperbolic groups have stable rank 1. For motivation and more examples see [Dykema, Kenneth J.; de la Harpe, Pierre Some groups whose reduced $C^*$-algebras have stable rank one. J. Math. Pures Appl. (9) 78 (1999), no. 6, 591–608.]

Here is a basic question, which I do not know how to answer.

Question. Suppose that $Q$ is a group of stable rank 1 and let $G$ be an extension $$ 1\to K\to G\to Q\to 1, $$ where $K$ is finite. Is it true that $G$ also has stable rank 1?

For example, if one wants to show that all non-elementary hyperbolic groups (not necessarily torsion free) have stable rank 1, then the affirmative answer to the above question would be a natural first step.