Twisted loop transgression and higher Jandl gerbes over finite groupoids

Given a double cover $\pi: \mathcal{G} \rightarrow \hat{\mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $\tau_{\pi}$ and $\tau_{\pi}^{ref}$, thereby associating to Jandl $n$-gerbe $\hat{\lambda}$ on $\hat{\mathcal{G}}$ $(n-1)$-gerbe $\tau_{\pi}(\hat{\lambda})$ the quotient groupoid $\mathcal{G}$ an ordinary $\tau^{ref}_{\pi}(\hat{\lambda})$ unoriented $\mathcal{G}$. For $n =1,2$, interpret character theory (resp. centre) category Real $\hat{\lambda}$-twisted $n$-vector bundles over in terms flat sections $(n-1)$-vector bundle associated $\tau_{\pi}^{ref}(\hat{\lambda})$ $\tau_{\pi}(\hat{\lambda})$). We relate our results versions Drinfeld doubles pointed fusion categories discrete torsion orientifold string M-theory.