  The quantization of chiral fermions on a 3-manifold in an external gauge
potential is known to lead to an abelian extension of the gauge group. In this
article we concentrate on the case of $\Omega^3 G$ of based smooth maps on a
3-sphere taking values in a compact Lie group $G.$ There is a crossed module
constructed from an abelian extension $\widehat{\Omega^3 G}$ of this group and
a group of automorphims acting on it as explained in a recent article by
Mickelson and Niemim\"aki. We shall construct a representation of this crossed
module in terms of a repesentation of $\widehat{\Omega^3 G}$ on a space of
functions of gauge potentials with values in a fermionic Fock space and a
representation of the automorphism group of $\widehat{\Omega^3 G}$ as outer
automorphisms of the canonical anticommutation relations algebra in the Fock
space.
