Multiples of a subvariety in an algebraic group

I will explain short proofs of the following, which extend results of Bogomolov and Tschinkel. Let $X$ be a subgroup of a commutative algebraic group $G$ over the algebraic closure $k$ of a finite field such that $X$ generates $G$. Then $\bigcup_{\phi \in \operatorname{End} G} \phi(X(k)) =G(k)$. If $G$ is semiabelian, then we have the stronger conclusion $\bigcup_{n \ge 1} n X(k) = G(k)$.