If $A$ is an excellent commutative ring and $G$ is a finite group of automorphisms of $A,$ is the invariant subring $A^G$ still excellent? I think this is false -- because if not it would probably be written in EGA IV, or in the recent Astérisque volume about Gabber's works on uniformisation and étale cohomology. But if anybody has a counter-example...

In fact, in the case I'm interested in, $A$ is not «any» excellent ring but an affinoid algebra over a non-Archimedean, complete field. Does one know something about $A^G$? I doubt that it is automatically affinoid -- if yes, it would be excellent.