\simeq \pi'$. Since not all cuspidal $\pi, \pi'$ come from Artin representations, the $\frac 57$ bound is not even conjecturally sufficient for general $\pi, \pi'$. However, it seems reasonable to think that coincidences of a large fraction of Euler factors only happen for essentially algebraic reasons, so the density bounds are likely to be the same in both the Artin and automorphic cases.

Acknowledgements {#acknowledgements .unnumbered}
----------------

We thank a referee for pointing out an error in an earlier version. The first author was partially supported by a Simons Collaboration Grant. The second author was supported by Forschungskredit grant K-71116-01-01 of the University of Zürich and partially supported by grant SNF PP00P2-138906 of the Swiss National Foundation. This work began when the second author visited the first at the University of Oklahoma. The second author would like to thank the first author as well as the mathematics department of the University of Oklahoma for their hospitality.

Notation and Background
=======================

Throughout, $G$, $H$ and $A$ will denote finite groups, and $A$ will be abelian. Denote by $Z(G)$ the center of $G$.

If $G$ and $N$ are groups, by a (group) extension of $G$ by $N$ we mean a group $H$ 