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The quantization of chiral fermions on a 3-manifold in an external gaugepotential is known to lead to an abelian extension of the gauge group. In thisarticle we concentrate on the case of $\Omega^3 G$ of based smooth maps on a3-sphere taking values in a compact Lie group $G.$ There is a crossed moduleconstructed from an abelian extension $\widehat{\Omega^3 G}$ of this group anda group of automorphims acting on it, \cite{MN}. We shall construct arepresentation of this crossed module in terms of a repesentation of$\widehat{\Omega^3 G}$ on a space of functions of gauge potentials with valuesin a fermionic Fock space and a representation of the automorphism group of$\widehat{\Omega^3 G}$ as outer automorphisms of the canonical anticommutationrelations algebra in the Fock space.

A crossed module representation of a $2$-group constructed from the $3$-loop group $\Omega^3 G$