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