\] Let $\psi$ be an Arthur parameter for ${}^{\vee}{G}^{\Gamma}$ satisfying points $\mathrm{(AJ0)}$, $\mathrm{(AJ0)}$ and $\mathrm{(AJ0)}$, above. Let $L^{\Gamma}$ be the extended group defined in (\[eq:extendedgroupL\]), and let $\left({}^{\vee}{L}^{\Gamma},\mathcal{S}_L\right)$ be the $E$-group defined in (\[eq:e-groupL\]). Write $\psi_{{L}}$ for the unique (up to conjugation) Arthur parameter of ${}^{\vee}{L}^{\Gamma}$ satisfying $\psi=\iota_{L,G}\circ\psi_{L}$. Let $\iota:X\left({}^{\vee}{L}^{\Gamma}\right)\rightarrow X\left({}^{\vee}{G}^{\Gamma}\right)$