  Let $X/\mathbb{F}_{q}$ be a smooth geometrically connected variety. Inspired
by work of Corlette-Simpson over $\mathbb{C}$, we formulate a conjecture that
absolutely irreducible rank 2 local systems with infinite monodromy on $X$ come
from families of abelian varieties. When $X$ is a projective variety, we prove
a Lefschetz-style theorem for abelian schemes of $\text{GL}_2$-type on $X$,
modeled after a theorem of Simpson. If one assumes a strong form of Deligne's
($p$-adic) \emph{companions conjecture} from Weil II, this implies that our
conjecture for projective varieties also reduces to the case of projective
curves. We also answer affirmitavely a question of Grothendieck on extending
abelian schemes via their $p$-divisible groups.
