The title says it all, I'm wondering what number of different (not necessarily regular) $4$-simplices can be inscribed into a $600$-cell modulo the full symmetry group of the latter (the vertices of each $4$-simplex should be among the vertices of the $600$-cell). Degenerate $4$-simplices (i.e. those lying in a single hyperplane) don't count.
I would probably try to approach this using the lemma which is not Burnside's. I originally used it to count the number of triangles in an $n$-gon up to rotations and reflections, which comes out to $\big[\frac{n^2}{12}\big]$, and the number of tetrahedra in an icosahedron up to rotations ($10$) and up to rotations plus reflections ($8$) (I think at least; the icosahedron is pretty nasty to look at). The problem with this approach is that it requires some visual intuition about the geometry and this is doesn't really work in $4D$. I have no idea how to even begin.