Frobenius--Schur indicators for twisted Real representation theory and two dimensional unoriented topological field theory

We construct a two dimensional unoriented open/closed topological field theory from a finite graded group $\pi :\widehat{G}\twoheadrightarrow \{1,-1\}$, a $\pi$-twisted $2$-cocycle $\widehat{\theta }$ on $B\widehat{G}$ and a character $\lambda :\widehat{G}\to U(1)$. The underlying oriented theory is a twisted Dijkgraaf-Witten theory. The construction is based in the $(\widehat{G},\widehat{\theta },\lambda )$-twisted Real representation theory of $\ker \pi$. In particular, twisted Real representations are boundary conditions and the generalized Frobenius-Schur element is its crosscap state.