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}