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This excellent question here does not seem to have an acceptable answer. I have had precisely the same question recently.

Namely, the way BRST quantization is usually presented relies on some kind of continuity in the gauge transform. For example, lots of treatments proceed as follows. The standard path integral $\int \mathcal{D} A e^{-S[A]}$ is nonsensical and should be replaced by a path integral over gauge orbits. This can be done by inserting a $\delta(F[A])$ into the path integral (where $F[A]$ is gauge fixing), but that requires a normalizing $\det \delta_\alpha F[A]$ factor. The latter now assumes continuity. For example, in a caricature of a simple $U(1)$, set $F[A] = A_0$, $\delta_\alpha F[A] = \frac{\delta}{\delta \alpha(x)}(A_0 + d\alpha(x')) = \partial \delta(x-x')$.

Is there an extension of BRST to where the gauge group is discrete?

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