Cuspidal $\mathrm{\ell }$-modular representations of ${\mathrm{G}\mathrm{L}}_n(F)$ distinguished by a Galois involution

Let $F\mathrm{/}{F}_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\mathrm{\ne }2$ with Galois automorphism $\sigma$, and let $R$ be an algebraically closed field of characteristic $\mathrm{\ell }\mathrm{\notin }\{0,p\}$. We reduce the classification of ${\mathrm{G}\mathrm{L}}_n(F_0)$-distinguished cuspidal $R$-representations of ${\mathrm{G}\mathrm{L}}_n(F)$ to the level $0$ setting. Moreover, under a parity condition, we give necessary conditions for a $\sigma$-selfdual cuspidal $R$-representation to be distinguished. Finally, we classify the distinguished cuspidal ${\bar{\mathbb{F}}}_{\mathrm{\ell }}$-representations of ${\mathrm{G}\mathrm{L}}_n(F)$ having a distinguished cuspidal lift to ${\bar{\mathbb{Q}}}_{\mathrm{\ell }}$.