A criterion for virtual global generation

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let $F_X$ denote the absolute Frobenius morphism of $X$ when the characteristic of $k$ is positive. A vector bundle over $X$ is called virtually globally generated if its pull back, by some finite morphism to $X$ from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of $k$ is positive, a vector bundle $E$ over $X$ is virtually globally generated if and only if ${\left(F_X^m\right)}^{*}E\cong E_a\oplus E_f$ for some $m$, where $E_a$ is some ample vector bundle and $E_f$ is some finite vector bundle over $X$ (either of $E_a$ and $E_f$ are allowed to be zero). If the characteristic of $k$ is zero, a vector bundle $E$ over $X$ is virtually globally generated if and only if $E$ is a direct sum of an ample vector bundle and a finite vector bundle over $X$ (either of them are allowed to be zero).