Let G be a central extension of a finite group H by $Z/2$. I need an explicit description of the differentials $d_2$ and $d_3$ in the Lyndon-Hochschild-Serre spectral sequence which converges to the cohomology of G with coefficients in either $Z/2$ or $U(1)$. It appears that $d_2$ is related to the cup product with the extension class, while $d_3$ is related to the Bockstein of the extension class, but I could not find it in any standard books on cohomology of groups.
Ok, maybe not in a standard book, but a classic anyway: the description of the differentials $d_2$ and $d_3$ can be found in the paper
- J.-P. Serre. Cohomologie modulo 2 des complexes d'Eilenberg–MacLane. Comment. Math. Helv. 27 (1953), 192-232. (MR)
As expected in the question, $d_2$ is the cup product with the extension class and $d_3$ is related to the Bockstein of the extension class via the product structure on the $E_3$-page. I actually found this in the introduction of the following paper
- I. Leary. A differential in the Lyndon–Hochschild–Serre spectral sequence. JPAA 88 (1993), 155-168. (MR)
which gives some more discussion and identifies the $d_4$-differential.