Automorphisms and twisted forms of rings of invariants

Let $G\subset G{L}_n(k)$ be a finite subgroup and $k[x_1,\dots ,x_n{]}^G\subset k[x_1,\dots ,x_n]$ its ring of invariants. We show that, in many cases, the automorphism group of $k[x_1,\dots ,x_n{]}^G$ is $k^{\times }$. Version 2: Incorporates parts of arXiv:2210.16265.