Let $X$ be an arbitrary scheme. A quasi coherent sheaf $\cal F$ is said to be injective if $Hom_{ O_X}(-, \cal F)$ is exact. We can also regard a quasi coherent sheaf $\cal G$ on $X$ such that for all open subset $U$ of $X$, $\cal G(U)$ is an injective $\cal O_X$-module. So we can ask a question that

1) Is there any relation between these two kind of sheaves?

2) Which conditions on $X$ (or on $\cal F$) are needed to regard the first kind of these sheaves ($\cal F$) equivalent to the second one?