  The generalized wreath product of permutation groups is introduced. By means
of it we study the schurity problem for S-rings over a cyclic group $G$ and the
automorphism groups of them. Criteria for the schurity and non-schurity of the
generalized wreath product of two such S-rings are obtained. As a byproduct of
the developed theory we prove that $G$ is a Schur group whenever the total
number $\Omega(n)$ of prime factors of the integer $n=|G|$ is at most 3.
Moreover, we describe the structure of a non-schurian S-ring over $G$ when
$\Omega(n)=4$. The latter result implies in particular that if $n=p^3q$ where
$p$ and $q$ are primes, then $G$ is a Schur group.
