Let $A$ be an abelian variety defined over a finite field $k$, and $End(A)$ be it ring of endomorphisms defined over an algebraic closure $\overline{k}$ of $k$. Suppose that for an integer $M$ coprime with the characteristic of $k$ there exist an endomorphism $\phi\in End(A)$ such that

$M=(1+F-2F^2)\circ \phi$

where $F$ is the Frobenius endomorphism.

Do we have that $\phi$ is defined over $k$? If the answer is yes, how can I proove it?