Given an extension of finite groups \(1\to H\to G @>\pi>> K\to 1\) and a prime \(p\) dividing the order of \(G\), let \(|A_p(K)|\) denote the geometric realization of the poset of elementary abelian \(p\)-subgroups of \(K\) and, for any \(i\)-simplex \(\sigma_i\), denote its stabilizer by \(K_{\sigma_i}\) and its orbit representative by \([\sigma_i]\). The main result of the paper measures how far \(H^*(G)\) differs from \(H^*(H)^K\); more precisely, it establishes a cohomology isomorphism with modulo \(p\) coefficients between \[ H^*(G)\oplus\bigl(\bigoplus_{[\sigma_i],\;i\text{ odd}} H^*(\pi^{-1}(K_{\sigma_i}))\bigr)\oplus\bigl(\bigoplus_{[\sigma_i],\;i\text{ even}} H^*(H)^{K_{\sigma_i}}\bigr) \] and \[ H^*(H) ^K\oplus\bigl(\bigoplus_{[\sigma_i],\;i\text{ even}}H^*(\pi^{-1}(K_{\sigma_i}))\bigr)\oplus\bigl(\bigoplus_{[\sigma_i],\;i\text{ odd}} H^*(H)^{K_{\sigma_i}}\bigr) \] where the \([\sigma_i]\) run over the simplices of \(|A_p(K)|/K\). This is then applied to group extensions of the above kind with \(H\cong(\mathbb{Z}/p)^n\). These arise in the study of the cohomology of some sporadic groups. For example, an application of the main result yields a decomposition of the mod 2 cohomology of certain sporadic simple groups exhibiting the contributions of various rings of invariants.