According to this page the answer is no, you can't inscribe an octahedron in an icosahedron so the vertices coincide. You can do it, however, to make the faces coincide.
If group theory doesn't worry you, the wikipedia page on icosahedral symmetry has information about the symmetry subgroups of the icosahedron, and confirms that you can inscribe tetrahedra and cubes in icosahedra so the vertices coincide.