suppose we have an exact sequence of locally free sheaves on a smooth variety: $$ 0\rightarrow E \rightarrow F \stackrel{f}{\rightarrow} G \rightarrow 0. $$ On an open subset $U\subset X$, this splits. Suppose $H$ is a proper subsheaf of $G$. Then on $U$, regard $H$ as a subsheaf of $F$. take its coherent extension $H'\subset f^{-1}(H) $ on $X$. It is torsion free. Then the kernel of $H'\rightarrow H$ on $X$ is subsheaf of $E$ and supported on $X-U$. hence the kernel is zero since $E$ is locally free.

Is the cokernel of $H'\rightarrow H$ supported in codimension two subset (it is a torsion sheaf).

suppose we have an exact sequence of locally free sheaves on a smooth variety: $$ 0\rightarrow E \rightarrow F \stackrel{f}{\rightarrow} G \rightarrow 0. $$ On an open subset $U\subset X$, this splits. Suppose $H$ is a proper subsheaf of $G$. Then on $U$, regard $H$ as a subsheaf of $F$. take its coherent extension $H'\subset f^{-1}(H) $ on $X$. It is torsion free. Then the kernel of $H'\rightarrow H$ on $X$ is subsheaf of $E$ and supported on $X-U$. hence the kernel is zero since $E$ is locally free.

Assume that $H$ is reflexive. Is it correct that the cokernel of $H'\rightarrow H$ is supported in codimension two subset (it is a torsion sheaf). ?

(take double dual and use that $H'\subset H'^{**}$ is isomorphism outside codim two subset)

suppose we have an exact sequence of locally free sheaves on a smooth variety: $$ 0\rightarrow E \rightarrow F \stackrel{f}{\rightarrow} G \rightarrow 0. $$ On an open subset $U\subset X$, this splits. Suppose $H$ is a proper subsheaf of $G$. Then on $U$, regard $H$ as a subsheaf of $F$. take its coherent extension $H'\subset f^{-1}(H) $ on $X$. Then the kernel of $H'\rightarrow H$ on $X$ is subsheaf of $E$ and supported on $X-U$. hence the kernel is zero since $E$ is locally free.

Is it possible to make the cokernel of $H'\rightarrow H$ zero (it is a torsion sheaf), without increasing the rank of $H'$.

suppose we have an exact sequence of locally free sheaves on a smooth variety: $$ 0\rightarrow E \rightarrow F \stackrel{f}{\rightarrow} G \rightarrow 0. $$ On an open subset $U\subset X$, this splits. Suppose $H$ is a proper subsheaf of $G$. Then on $U$, regard $H$ as a subsheaf of $F$. take its coherent extension $H'\subset f^{-1}(H) $ on $X$. It is torsion free. Then the kernel of $H'\rightarrow H$ on $X$ is subsheaf of $E$ and supported on $X-U$. hence the kernel is zero since $E$ is locally free.

Is the cokernel of $H'\rightarrow H$ supported in codimension two subset (it is a torsion sheaf).