These are the Kepler-Poinsot (or "Star") polyhedra.
How many of them share the same vertex arrangement as an icosahedron?
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They all have 1 2 vertices and share the same vertex arrangement as the icosahedron except for the great stellated dodecahedron which has 2 0 vertices that share the same vertex arrangement as the dodecahedron.
So the answer is 3 of them share the same vertex arrangement as the icosahedron