Are There Highly Symmetric NP- or P-complete Languages?
For NP, this seems hard to construct. In particular, if you can also sample (nearly) uniform elements from your group - which is true for many natural ways of constructing groups - then if an NP-complete language has a poly-time group action with few orbits, PH collapses. For, with this additional assumption about sampleability, the standard $\mathsf{coAM}$ protocol for Graph Isomorphism also works for testing whether two strings are in the same $G_n$-orbit. We then would have $\mathsf{NP} \subseteq \mathsf{coAM/poly} = \mathsf{coNP/poly}$, so PH collapses to $\mathsf{ZPP}^{\mathsf{NP}}$. So, to avoid collapsing PH, any such construction for NP would need the groups to not have an efficient nearly-uniform sampler.