We study K-theoretic Gromov--Witten invariants of projective hypersurfaces using a virtual localization formula under finite group actions. In particular, it provides all K-theoretic Gromov--Witten invariants of the quintic threefold modulo 41, up to genus 19 and degree 40. As an illustration, we give an instance in genus one and degree one. Applying the same idea to a K-theoretic version of FJRW theory, we determine it modulo 41 for the quintic polynomial with minimal group and narrow insertions, in every genus.