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It is well-known that the value of the Frobenius-Schur indicator $|G|^{-1}\sum_{g\in G} \chi(g^2)=\pm1$ of a real irreducible representation of a finitegroup $G$ determines which of the two types of real representations it belongsto, i.e. whether it is strictly real or quaternionic. We study the extension tothe case when a homomorphism $\varphi:G\to \mathbb{Z}/2\mathbb{Z}$ gives thegroup algebra $\mathbb{C}[G]$ the structure of a superalgebra. Namely, weconstruct of a super version of the Frobenius-Schur indicator whose value for areal irreducible super representation is an eighth root of unity,distinguishing which of the eight types of irreducible real superrepresentations described in [Wall1964] it belongs to. We also discuss itssignificance in the context of two-dimensional finite-group gauge theories onpin$^-$ surfaces.

The super Frobenius-Schur indicator and finite group gauge theories on pin$^-$ surfaces