Direct sums, submodules and simple tops

Question

Let $A$ be a finite dimensional algebra over some field.

Suppose $X_1$ and $X_2$ are $A$-modules, both with simple top (that is, $X_i/ (\operatorname{Rad} X_i)$ is simple for $i=1,2$). Let $K$ be a submodule of $X_1 \oplus X_2$, such that $(X_1 \oplus X_2)/K$ has simple top.

Does it follow that $X_1 \subseteq K$ or $X_2 \subseteq K$?

Answer 1

No: Take any field $F$. Let $A=X_1=X_2=F$ and $K=\{(x,x)\in X_1 \oplus X_2 : x \in F\}$. Then $X_1$, $X_2$, and $X_1\oplus X_2 /K$ are all simple, so have simple tops, but $K$ is not contained in either $X_1$ or $X_2$.