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A quite straightforward way of seeing that the fundamental group is binary icosahedral is to solve exercise 4.4.17 in page 252 of Thurston's book 3-D Geom & Topo (the published one in 1997). When doing this exercise, one should notice two things: 1. on $S^2$, each of the $5$ angles of a pentagon face of a dodecahedron is $120$ degrees, in contrast to $108$ degrees for a pentagon on plane - this is useful when you try to create a coordinate system on the circle bundle $UTS^2$; 2. the "fundamental domain" Thurston created here does not look like a dodecahedron in the metric sense - it has only $5$ side faces, a pentagon top face and a pentagon bottom face; however other "fundamental domains" will mark a line on each of the side faces and therefore each side face splits into two "pentagons"... With these in mind, exercise 4.4.17 will be very pleasant to work out.