Lifting an injective resolution of a flat sheaf

Question

Let $X$ be a projective variety over a field $k$, and $F$ a coherent sheaf on $X$. Choose an injective resolution $F\to I^\bullet$ on $X$. Let $A$ be an Artin local $k$-algebra, and $F_A$ an $A$-flat coherent sheaf on $X_A=X\times\operatorname{Spec}A$ such that $F_A|_{X}\cong F$, i.e., a deformation of $F$ over $A$. Is there a complex $( A\otimes_k I^\bullet,d_A^\bullet)$ on $X_A$ with $H^0(A\otimes I^\bullet,d_A^\bullet)=F_A$ and restricts to the given resolution $F\to I^\bullet$? Thanks for help.