I'm looking for an example of the following situation:

- A group $G$ generated by finite subgroups $H$ and $K$,
- a non-trivial 3-cocycle $\omega \in H^3(G, \mathbb{k}^\times)$

such that

- the restrictions of $\omega$ to a 3-cocycle on each of $H$ or $K$ is a coboundary.

If such an example is possible with at least one of $H^2(H, \mathbb{k}^\times)$ and $H^2(K, \mathbb{k}^\times)$ non-trivial as well that would be even better