  Lattice regularizations are pivotal in the non-perturbative quantization of
gauge field theories. Wilson's proposal to employ group-valued link fields
simplifies the regularization of gauge fields in principal fiber bundles,
preserving gauge symmetry within the discretized lattice theory. Maintaining
gauge symmetry is desirable as its violation can introduce unwanted degrees of
freedom. However, not all theories with gauge symmetries admit gauge-invariant
lattice regularizations, as observed in general relativity where the
diffeomorphism group serves as the gauge symmetry. In such cases, gauge
symmetry-breaking regularizations become necessary. In this paper, we argue
that a broken lattice gauge symmetry is acceptable as long as gauge symmetry is
restored in the continuum limit. We propose a method to construct the continuum
limit for a class of lattice-regularized Hamiltonian field theories, where the
regularization breaks the Lie algebra of first-class constraints. Additionally,
we offer an approach to represent the exact gauge group on the Hilbert space of
the continuum theory. The considered class of theories is limited to those with
first-class constraints linear in momenta, excluding the entire gauge group of
general relativity but encompassing its subgroup of spatial diffeomorphisms. We
discuss potential techniques for extending this quantization to the full gauge
group.
