Let $G$ be a finite group. Let $x,y \in Z^2(G,\mathbb{Z}_2)$ be 2-cocycles. Find $a \in C^2(G,\mathbb{Z}_2)$ such that

\begin{align} x \cup_1 y = \delta a. \end{align}

Is there a general solution? Is it possible to know when a solution exists?

Where: \begin{align} [x \cup_1 y](g,h,k) &= x(gh,k)y(g,h) + x(g,hk)y(h,k)\\ \delta a &= a(g,h)+a(gh,k)-a(g,hk)-a(h,k) \end{align}