Orbit recovery from invariants of low degree in representations of finite groups

Motivated by applications to equivariant neural networks and cryo-electron microscopy we consider the problem of recovering the generic orbit in a representation of a finite group from invariants of low degree. The main result proved here is that invariants of degree at most three separate generic orbits in the regular representation of a finite group defined over any infinite field. This answers a question posed in a 2023 ACHA paper of Bandeira et. al. We also discuss this problem for subregular representations of the dihedral and symmetric groups.