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 a Jandl $n$-gerbe $hat{lambda}$ on $hat{mathcal{G}}$ a Jandl $(n-1)$-gerbe $tau_{pi}(hat{lambda})$ on the quotient loop groupoid of $mathcal{G}$ and an ordinary $(n-1)$-gerbe $tau^{ref}_{pi}(hat{lambda})$ on the unoriented quotient loop groupoid of $mathcal{G}$. For $n =1,2$, we interpret the character theory (resp. centre) of the category of Real $hat{lambda}$-twisted $n$-vector bundles over $hat{mathcal{G}}$ in terms of flat sections of the $(n-1)$-vector bundle associated to $tau_{pi}^{ref}(hat{lambda})$ (resp. the Real $(n-1)$-vector bundle associated to $tau_{pi}(hat{lambda})$). We relate our results to Re
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