Let $M$ be a module over a polynomial ring $k[x_1, \dots, x_n]$ and let $E/k$ be a finite field extension. In particular, $M \otimes_k E$ is an $E[x_1, \dots, x_n]$-module. Is it true that every subquotient of $M \otimes_k E$ is isomorphic to $M' \otimes_k E$ for $M'$ a subquotient of $M$? Recall that a subquotient of a module $N$ is a submodule of a quotient of $N$.
I feel like this should be false but I haven't been able to find a counterexample.