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Canonical Homomorphism $G_{2\times 2} \to \mathrm{PGL}(3,2)$?

Does there exist a natural geometric realization of the $2\times 2$ Rubiks cube group $G_{2\times 2}$ that induces a canonical homomorphism $G_{2\times2} \to \mathrm{PGL}(3,2)$?