  In this paper, we study the isomorphism problem for central extensions. More
precisely, in some new situations, we provide necessary and sufficient
conditions for two central extensions to be isomorphic. We investigate the case
when the quotient group is simple or purely non-abelian. Furthermore, we
characterize isomorphisms leaving the quotient group invariant. Finally, we
deal with isomorphisms of central extensions where the kernel group and the
quotient group are isomorphic.
