  Let $G_{2}$ be a group which acts trivially on an abelian group $G_{1}$. As
is well known, each perturbed direct product of $G_{1}$ and $G_{2}$ under a
2-cocycle $\varepsilon\in Z^{2}(G_{2},G_{1})$ determines a central extension of
$G_{1}$ by $G_{2}$. The purpose of this paper is to study perturbed direct
products of groups and to decide in some cases how the isomorphism of these
groups can be decided. Furthermore, we show that the study of the isomorphism
of perturbed direct products of an abelian torsion group and a finite group is
reduced to the study of the isomorphism of $p$-subgroups. We characterize such
isomorphisms in various situations with some assumptions on the quotient group.
