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 



A group 3-cocycle, trivial on a pair of generating subgroups?
If $\mathbb{k}^\times$ is the multiplicative group of some field, the following works with $\mathbb{k}=\mathbb{F}\_3$: 


The Quaternion group $Q\_8$ is generated by two cyclic subgroups $H,K$ of order 4 and 
$$H^\ast(H;\mathbb{F}\_2)\cong H^\ast(K;\mathbb{F}\_2)\cong \mathbb{F}\_2[a,b]/(a^2)\; ,\quad|a|=1, |b|=2$$ 
$$H^\ast(Q\_8;\mathbb{F}\_2) = \mathbb{F}\_2[x,y,z]/(xy,x^3-y^3)\;,\quad |x|=|y|=1,|z|=4$$
Thus $x$ restricts on $H,K$ to some class $c$ having $c^2=0$ and consequently $x^3\neq 0$ restricts to zero on $H,K$. 


