Clearly, my question can be asked more generally, but suppose for simplicity that $X$ is a smooth surface and that $0 \to E \to F \to K \to 0 $ is exact with $E ,F$ rank two bundles on $X$ and $K$ a line bundle on a (smooth?) divisor $D \subset X$. Can one give information on the cokernel of the injection $Sym^n(E) \to Sym^n(F) $ ? For example is this cokernel some obvious sum of linear algebra constructions (various symmetric powers) of $E$ , $F$, and $K$ ? I can only say that I have thought about this question for a while and no answer has occurred to me.