How to get the nth-order tesselation of a polyhedron in Mathematica?

Question

I'd like to get the n-th order tesselation of a polyhedron like in the drawings on Wikipedia.

https://en.wikipedia.org/wiki/Geodesic_polyhedron#/media/File:Geodesic_icosahedral_polyhedron_example.png https://en.wikipedia.org/wiki/Geodesic_polyhedron#/media/File:Geodesic_icosahedral_polyhedron_example2.png https://en.wikipedia.org/wiki/Geodesic_polyhedron#/media/File:Geodesic_icosahedral_polyhedron_example5.png

Mathematica Geodesate[] yields a subdivision / tesselation but I don't want the projection onto the sphere. In the above pictures the second last is my goal.

My goal is to produce a generalized star polyhedron.

Any help is appreciated.