Is there a clear and correct reference for full icosahedral symmetry that includes a presentation and its action on vertices, edges, and faces?
The Wikipedia article for icoshedral symmetry gives the following presentation for the full icosahedral group:
$$I_h: \langle s,t\mid s^3(st)^{-2}, t^5(st)^{-2}\rangle$$
This of course is the presentation for $\operatorname{SL}(2,5)$ as it is a 2-fold covering of the 2-3-5 group $\langle s,t \mid (st)^2 = s^3 = t^5 = 1\rangle$. However, the article is quite clear about this being the wrong group.
This confusion has been in the article since Jan 2006.