Summary: The representations of a group of gauge automorphisms of the canonical commutation or anticommutation relations which appear on the Hilbert spaces of isometries \(H_\varrho\) implementing quasi-free endomorphisms \(\varrho\) on Fock space are studied. Such a representation, which characterizes the “charge” of a \(\varrho\) in local quantum field theory, is determined by the Fock space structure of \(H_\varrho\) itself: Together with a “basic” representation of the group, all higher symmetric or antisymmetric tensor powers thereof also appear. Hence \(\varrho\) 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 implement ability of quasi-free endomorphisms are compatible with each other.