On the cohomology of split extensions of finite groups

Let $G=H\rtimes Q$ be a split extension of finite groups. A theorem of Charlap and Vasquez gives an explicit description of the differentials $d_2$ in the Lyndon-Hochschild-Serre spectral sequence of the extension with coefficients in a field $k$. We generalize this to give an explicit description of all the $d_r$ ($r\geq 2$) in this case. The generalization is obtained by associating to the group extension a new twisting cochain, which takes values in the $kG$-endomorphism algebra of the minimal $kH$-projective resolution induced from $H$ to $G$. This twisting cochain not only determines the differentials, but also allows one to construct an explicit $kG$-projective resolution of $k$.