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Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be a finiteGalois extension of $F$ with degree of extension $l$, where $l$ and $p$ aredistinct odd primes. Let $\pi_F$ be an integral, $l$-adic genericrepresentation of ${\rm GL}_3(F)$, and let $\pi_E$ be the base change liftingof $\pi_F$ to ${\rm GL}_3(E)$. Let $J_l(\pi_F)$ (resp. $J_l(\pi_E)$) be theunique generic sub-quotient of the mod-$l$ reduction of $\pi_F$ (resp.$\pi_E$). In this article, using the local converse theorem over local Artinian$\overline{\mathbb{F}}_l$-algebras, we prove that the Frobenius twist of$J_l(\pi_F)$ is isomorphic to the Tate cohomology group $\widehat{H}^0({\rmGal}(E/F),J_l(\pi_E))$. The result of this article removes the hypothesis thatthe prime $l$ does not divide the pro-order of ${\rm GL}_2(F)$.

Tate cohomology and local base change of generic representations of ${\rm GL}_3$ -- non-banal case