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 LyndonHochschildSerre 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.
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1$\begingroup$ The paper by Johannes Heubschmann mentioned in Andy Putman's answer to this question mathoverflow.net/questions/590/… might have some useful information. You could also try using Kudo's Transgression theorem to make sense of your claim about $d_3$. $\endgroup$– Mark GrantMar 22, 2016 at 7:11

1$\begingroup$ We've done lots of computations for $H=Z/2$, see pages 1417 in arxiv.org/abs/hepth/0701071. There we used a socalled 7term exact sequence in order to determine $d_2$. Maybe that can help, good luck! $\endgroup$– Konrad WaldorfMar 22, 2016 at 15:38
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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), 192232. (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), 155168. (MR)
which gives some more discussion and identifies the $d_4$differential.