Noether's problem is a famous problem in invariant theory, introduced in the 1910's by Emmy Noether in relation to the inverse Galois problem. It is as follows:
Noether's Problem: Let $F=k(x_1,\dotsc,x_n)$ be a purely transcendental extension of a base field $k$ and $G<S_n$ a group of permutations acting transitively on the set of variables. Is $F^G$ a purely transcendental extension of $k$?
Now the problem can be generalized, to what I've seen people call the General Noether's Problem (I don't know if this terminology is standard):
General Noether's Problem: Let $F=k(x_1,\dotsc,x_n)$ and $G$ be any finite group of $k$-fields automorphisms of $F$. When is $F^G$ purely transcendental over $k$?
I know two cases of General Noether's Problem that are much studied: the case $G<\operatorname{GL}_n(k)$ (Linear Noether's Problem), and the case $G<\operatorname{GL}_n(\mathbb{Z})$ acts by monomial automorphisms, that is, if $g \in G$, $g=(a_{ij})$ then
$$ g.x_j=\prod_{i=1}^n x_i^{a_{ij}}, j=1,\dotsc,n. $$
I am interested in cases of positive solution to General Noether's Problem that are not cases of permutation, linear or monomial actions.
In case $F=k(x)$ of course any finite $G$ gives a positive solution, by Lüroth's Theorem, and if $F=k(x,y)$ and $k$ is algebraically closed of zero characteristic, the same holds due to Castelnuovo's Theorem.
What is known about positive solutions of General Noether's Problem besides these two particular cases?
