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 tow kind of sheaves?

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