  Let $G\subset GL_n(k)$ 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.
