  The group isomorphism problem asks whether two given groups are isomorphic or
not. Whereas the case where both groups are abelian is well understood and can
be solved efficiently, very little is known about the complexity of isomorphism
testing for nonabelian groups. In this paper we study this problem for a class
of groups corresponding to one of the simplest ways of constructing nonabelian
groups from abelian groups: the groups that are extensions of an abelian group
A by a cyclic group of order m. We present an efficient algorithm solving the
group isomorphism problem for all the groups of this class such that the order
of A is coprime with m. More precisely, our algorithm runs in time almost
linear in the orders of the input groups and works in the general setting where
the groups are given as black-boxes.
