Summary: A finite group \(G\) is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class \([c]\in H^2(G,\mathbb{C}^*)\) (\(G\) acts trivially on \(\mathbb{C}^*\)). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation-theoretical properties.
Suppose that a finite group \(Q\) acts on an Abelian group \(A\) so that there exists a bijective 1-cocycle \(\pi\in Z^1(Q,\check A)\), where \(\check A=\operatorname{Hom}(A,\mathbb{C}^*)\) is endowed with the diagonal \(Q\)-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in \(Z^2(G,\mathbb{C}^*)\), where \(G:=A\rtimes Q\). Hence, the semidirect product \(G\) is of central type.
In this paper, we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class \(\pi\in H^1(Q,\check A)\) as above, we construct non-degenerate classes \([c_\pi]\in H^2(G,\mathbb{C}^*)\) for certain extensions \(1\to A\to G\to Q\to 1\) which are not necessarily split. We thus strictly extend the above family of central type groups.