Differentials in the Lyndon-Hoschild-Serre Sequence for p=0

Question

I'm interested in whether there is a simple description of the differentials in the first column of the LHS spectral sequence (the column with $E_2^{0,q}=H^0(BK,H^q(BG))=H^q(BG)^K$ for a short exact sequence $$ 1 \to G \to H \to K \to 1. $$

I believe these should be much simpler to understand than the general differentials, since when the above sequence splits these differentials are all zero (unless I'm mistaken), which is certainly not the case for $p\geq 1$.

I would be satisfied with some formulas for small $q$ (say $\leq 6$ or so) such as $d_3(x) =$ the contraction of $x$ with the extension class $\omega$.

Answer 1

The differential on the E_2 page is given by cup product with (essentially) the class of the extension. See [André, Michel Le d2 de la suite spectrale en cohomologie des groupes. C. R. Acad. Sci. Paris 260 1965 2669–2671.]

(It is done there only for abelian kernels, but should work in general)

Answer 2

Johannes Huebschmann has written several papers on this topic. Here are three references:

Automorphisms of group extensions and differentials in the Lyndon-Hochschild-Serre spectral sequence. J. Algebra 72 (1981), no. 2, 296–334.

Group extensions, crossed pairs and an eight term exact sequence. J. Reine Angew. Math. 321 (1981), 150–172.

Sur les premières différentielles de la suite spectrale cohomologique d'une extension de groupes. (French. English summary) C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 15, A929–A931.

In these papers you will find descriptions of the differentials $d_2^{0,2}:E_2^{0,2}\to E_2^{2,1}$, $d_2^{0,1}:E_2^{0,1}\to E_2^{2,0}$ and $d_3^{0,2}:E_3^{0,2}\to E_3^{3,0}$, among others.