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?