  Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact
fields of residual characteristic $p\neq2$ with Galois automorphism $\sigma$,
and let $R$ be an algebraically closed field of characteristic
$\ell\notin\{0,p\}$. We reduce the classification of
$\mathrm{GL}_n(F_0)$-distinguished cuspidal $R$-representations of
$\mathrm{GL}_n(F)$ 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 $\overline{\mathbb{F}}_{\ell}$-representations of $\mathrm{GL}_n(F)$
having a distinguished cuspidal lift to $\overline{\mathbb{Q}}_\ell$.
