Given an affine morphism that is also an open immersion $i:U\hookrightarrow X$ of Noetherian schemes and a coherent $O_U$-module $M$, $i_*M$ is quasi-coherent such that $i^*i_*M\cong M$. This implies that there is some coherent sub-sheaf of $i_*M$ that restricts to $M$ on $U$. My question is, is it possible to construct a functorial assignment $F(M)$, such that $F(M)$ is a coherent sheaf on $X$ where $i^*F(M)\cong M$ and the functor $F$ is exact?
My second question is whether it is possible to extend a vector bundle to another vector bundle? (In the above paragraph can we choose the coherent sub-sheaf of $i_*M$ to be a vector bundle if we know that $M$ is a vector bundle.)