Let $A$ be an abelian variety over a finite field $k$. Let $V_\ell(A)$ be its $\ell$-adic Tate module.

We have a natural action of the absolute Galois group of $k$, and thus an action of the Frobenius automorphism of $\overline{k}$ on $V_\ell(A)$.

However, $A$ also has an endomorphism $F:A\to A$ which also acts on $V_\ell(A)$ (only as an endomorphism a priori).

Are these endomorphisms of $V_\ell(A)$ the same? I am sure they are, but why? I have seen this used many times, and I'm slightly confused about it...