Possible uses of the binary icosahedral group in grand unified theories

There are exactly three finite subgroups of SU(2) that act irreducibly in the spin 1 representation, namely the binary tetrahedral, binary octahedral and binary icosahedral groups. In previous papers I have shown how the binary tetrahedral group gives rise to all the necessary ingredients for a non-relativistic model of quantum mechanics and elementary particles, and how a modification of the binary octahedral group extends this to the ingredients of a relativistic model. Here I investigate the possibility that the binary icosahedral group might be related in a similar way to grand unified theories such as the Georgi--Glashow model, the Pati--Salam model, various $E_8$ models and perhaps even M-theory.