Let $G$ be a finite group, $H$ a subgroup of $G$, and $A$ an abelian group.
Assume given a 3-cocycle $\omega:G\times G\times G\rightarrow A$ such that $\omega|_{H\times H\times H}$ is coboundary, say $\omega|_{H\times H\times H}=df$ for some $f:H\times H\rightarrow A$. It is possible to extend $f$ to a map $f':G\times G\rightarrow A$, such that the 3-cocycle $\omega' = \omega + df'$ satisfies $$\omega'(g_1,g_2,g_3)=0$$ if either both $g_1$ and $g_2$, or both $g_2$ and $g_3$ are in $H$. (I don't know if this fact appears in the litterature? In any case, I have a proof.) Does this guarantee that $$\omega'(g_1,g_2,g_3)=0$$ if $g_1$ and $g_3$ are in $H$???
I am particularly interested in the case where $G$ is abelian!