The representations of a group of gauge automorphisms of the canonical commutation or anticommutation relations which appear on the Hilbert spaces of isometries $H_\rho$ implementing quasi-free endomorphisms $\rho$ on Fock space are studied. Such a representation, which characterizes the "charge" of $\rho$ in local quantum field theory, is determined by the Fock space structure of $H_\rho$ itself: Together with a "basic" representation of the group, all higher symmetric or antisymmetric tensor powers thereof also appear. Hence $\rho$ is reducible unless it is an automorphism. It is further shown by the example of the massless Dirac field in two dimensions that localization and implementability of quasi-free endomorphisms are compatible with each other.