Classes in $H^3(G; \mathbb{Z})$ that restrict to zero on abelian subgroups Let $G$ be a finite $p$-group. Is it possible to have a nonzero class in $H^3(G; \mathbb{Z})$ that restricts to zero in $H^3(A; \mathbb{Z})$ for every abelian subgroup $A \subset G$? If so, what is a simple example? 
 A: I don't know about in general, but what you're after is tied in to (integral) Essential Cohomology $Ess_\mathbb{Z}(G)$, as follows:
Let $G$ be a finite group. An element $x\in H^*(G)$ is called essential if it has trivial restriction to all proper subgroups of $G$; the set of such elements make up the essential ideal $Ess_\mathbb{Z}(G)$. If $G$ is not a $p$-group, then this essential ideal is 0. Thus the main focus has been on mod-$p$ cohomology, especially when it's an easier ring to work under. But some stuff can be said for $\mathbb{Z}$-coefficients. For an example, here is a computation I did a long time ago:
Take $G=\mathbb{Z}_2\times\mathbb{Z}_2$, so that $H^*(G,\mathbb{Z})\cong\mathbb{Z}[\alpha,\beta,\mu]/(2\alpha,2\beta,\alpha\beta^2+\alpha^2\beta-\mu^2)$ where $|\alpha|=|\beta|=2$ and $|\mu|=3$. Then $\text{res}^G_{\mathbb{Z}_2\times 1}\alpha=\text{res}^G_{1\times \mathbb{Z}_2}\beta=\eta$ and $\text{res}^G_{\mathbb{Z}_2\times 1}\beta=\text{res}^G_{1\times \mathbb{Z}_2}\alpha=0$, where $H^*(\mathbb{Z}_2,\mathbb{Z})\cong\mathbb{Z}[\eta]/(2\eta)$ with $|\eta|=2$.  For the final maximal subgroup $M=\lbrace (0,0),(1,1)\rbrace$ we have $\text{res}^G_M\alpha=\eta$ and $\text{res}^G_M\beta=\eta$, so $\text{res}^G_M\alpha\beta=\eta^2$ and $\text{res}^G_M(\alpha+\beta)=0$.
Now $\mu$ must restrict to a $3$-dimensional element on the maximal subgroups, which it cannot because the rings are generated by a $2$-dimensional element $\eta$.  Thus $Ess_\mathbb{Z}(\mathbb{Z}_2\times\mathbb{Z}_2)=(\mu)$.
