I recently worked through most of the proof of the rationality of the moduli of genus 3 curves, which seemed to have the following structure:

- Every nonhyperelliptic genus 3 curve is a smooth plane quartic.
- The plane quartics form a projective space.
- Apply GIT to this projective space and the $PGL(3)$ action.
- Prove that this quotient is rational.

I've seen somewhat similarly structured arguments before. So my question:

When is a GIT quotient rational?

In particular, are quotients of $\mathbb{P}^n$ by $PGL_k$ rational, under some reasonable hypotheses?

Are there any natural invariants that are preserved by quotients (again, with reasonable conditions, or of the above form)?