Consider a curve $X_0$ over an algebraically closed field $k$. Let $A$ be a local Artinian ring over $k$ and $X$ a scheme over $A$ with $X \otimes_A k = X_0$. For a sheaf $F$ on $X$ we denote by $F_0$ the sheaf $F \otimes_A k$ on $X_0$, in other words, it is the inverse image of $F$ by the canonical inclusion $X_0 \to X$.
Let $E$ be a vector bundle on $X$ and assume there is an exact sequence $$ 0 \to E'_0 \to E_0 \to E''_0 \to 0. $$
Q: Is there a lift of the above exact sequence to a sequence of vector bundles on $X$ with $E$ in the middle?