  Given a finite $\mathbb{Z}_2$-graded group $\hat{\mathsf{G}}$ with ungraded
subgroup $\mathsf{G}$ and a twisted cocycle $\hat{\lambda} \in Z^n(B
\hat{\mathsf{G}}; \mathsf{U}(1)_{\pi})$ which restricts to $\lambda \in Z^n(B
\mathsf{G}; \mathsf{U}(1))$, we construct a lift of $\lambda$-twisted
$\mathsf{G}$-Dijkgraaf--Witten theory to an unoriented topological quantum
field theory. Our construction uses a new class of homotopy field theories,
which we call orientation twisted. We also introduce an orientation twisted
variant of the orbifold procedure, which produces an unoriented topological
field theory from an orientation twisted $\mathsf{G}$-equivariant topological
field theory.
