A note on Serre's theorem in group cohomology

In 1987, Serre proved that if $G$ is a $p$-group which is not elementary abelian, then a product of Bocksteins of one dimensional classes is zero in the mod $p$ cohomology algebra of $G$, provided that the product includes at least one nontrivial class from each line in $H^1 (G, \mathbb{F}_p)$. For $p=2$, this gives that $(\sigma _G )^2 =0$, where $\sigma_G$ is the product of all nontrivial one dimensional classes in $H^1 (G, \mathbb{F}_2)$. In this note, we prove that if $G$ is a nonabelian $2$-group, then $\sigma_G$ is also zero.