If $G$ is a compact Lie group acting smoothly on $\mathbb R^n$ by orthogonal matrices, a result of G. Schwartz characteriezs the smooth $G$-invariant functions $f : \mathbb R^n \to \mathbb R$ as smooth functions of the $G$-invariant polynomials $P(\mathbb R^n)^G$. (See this MO post for a discussion.)
Here I wondered whether a Riemannian metric descends smoothly through a ramified covering map, which led to a similar problem:
Question. When $G$ is a finite group (more generally, a compact Lie group) with smooth orthogonal actions on $\mathbb R^n$ and $\mathbb R^m$, what is known about $G$-equivariant smooth maps $$f : \mathbb R^n \to \mathbb R^m ?$$
(Schwartz' theorem is the case $m = 1$.)