  A finite group $G$ is of central type (in the non-classical sense) if it
admits a non-degenerate cohomology class $[c]\in H^2(G,\C^*)$ ($G$ acts
trivially on $\C^*$). 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 $\pi\in Z^1(Q,\ach)$, where
$\ach=\rm{Hom}(A,\C^*)$ is endowed with the diagonal $Q$-action. Under this
assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate
2-cocycle in $Z^2(G,\C^*)$, 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
$[\pi]\in H^1(Q,\ach)$ as above, we construct non-degenerate classes
$[c_{\pi}]\in H^2(G,\C^*)$ 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.
