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We study constraints on the space of $d=2$ fermionic CFTs as a function ofnon-perturbative anomalies exhibited under a fermionic discrete symmetry group$G^f$, focusing our attention also on cases where $G^f$ is non-abelian orpresents a non-trivial twist of the $\mathbb{Z}^f_2$ subgroup. For the cases weselected, among our results we find that modular bootstrap consistency boundspredict the presence of relevant/marginal operators only for some groups andanomalies. From this point of view, the appearance in the analysis of severalkinks around irrelevant operators with $\Delta>2$ means that for fermionicsystems with increasingly larger symmetry groups modular bootstrap is able togive less constraining bounds than its bosonic counterpart. Within our analysiswe show how the anomaly constraints on fermionic CFTs can be effectivelyrecovered from the structure of the abelian subgroups of $G^f$. Finally, weextend the previous surgery description of bordism invariants that describe$3d$ abelian spin-TQFTs, in order to include the case of theories with$\mathrm{Spin}\text{-}\mathbb{Z}^f_{2^{l+1}}$ structures.

Anomalies of fermionic CFTs via cobordism and bootstrap