On non-connected simple linear groups with a free algebra of invariants

In this paper we study non-connected coregular (that is, with a free algebra of invariants) linear groups. A criterion for the coregularity of a semisimple group $G\subseteq \operatorname{GL}(V)$ is obtained in terms of the action of $G/G^0$ on the quotient variety $V/G^0$. A classification of connected non-coregular simple linear groups which admit a finite coregular extension is found and such extensions are described in each case.