In this talk the fixed point structure of gravity-fermion systems on a curved background spacetime will be discussed using the functional renormalization group equation. The effective average action is approximated by the Einstein-Hilbert action supplemented by a fermion kinetic term and a coupling of the fermion-bilinears to the spacetime curvature. The latter interaction is singled out based on a “smart truncation building principle”. The resulting renormalization group flow possesses two families of interacting renormalization group fixed points extending to any number of fermions. The first family exhibits an upper bound on the number of fermions for which the fixed points could provide a phenomenologically interesting high-energy completion via the asymptotic safety mechanism. The second family comes without such a bound. The inclusion of the non-minimal gravity-matter interaction is crucial for discriminating the two families. some insights into the origin of the strong regulator-dependence of the fixed point structure reported in earlier literature is also provided. The relation of our findings to studies of the same system based on a vertex expansion of the effective average action around a flat background spacetime will also be discussed.