There are 59 stellations of the icosahedron. The faces of the stellations lie in one of 20 planes through the faces of the original icosahedron. Each stellation consists of a set of facets subjected to connectivity constraints (no holes).
The combinatorial analysis of the icosahedral stellations is by H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie, The Fifty-Nine Icosahedra, New York: Springer-Verlag, 1982. (Originally published by Univ. of Toronto Press, 1938.)
The Mathematica code was originally developed for R. E. Maeder, The Mathematica Programmer, Boston: AP Professional, 1994.