Let $(Q, \mathfrak n, k)$ be a regular local ring. Let $I\subseteq \mathfrak n^2$ be an ideal, and fix a minimal generating set $\mathbb f= f_1,\cdots, f_n$ of $I$. The Koszul complex $E:= Q[e_1,...,e_n| \partial e_i=f_i]$ of $\mathbb f$ over $Q$ carries a natural DG-algebra structure. Let $M$ be a finitely generated $Q/I$-module, hence also a $Q$-module.

Let $F_M$ be a minimal $Q$-free resolution of $M$. Note that this resolution is bounded as $Q$ is regular. My question is: Does $F_M$ have a DG $E$-algebra structure? If this is not necessarily true, what if I also assumed $M\cong Q/J$ for some ideal $J$ containing $I$ ?