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We suggest a criterion under which for a nilpotent group of finite exponent$A$ and for an abelian group $B$ the variety $var(A \,Wr\, B)$ generated bytheir wreath product $A \,Wr\, B$ is equal to the product of varieties $var(A)$and $var(B)$ generated by $A$ and $B$. Namely the equality holds if and only ifeither the group $B$ is not of some non-zero exponent; or if $B$ is of anon-zero exponent $n$, and $B$ contains a subgroup isomorphic to $C_{d}^c\times C_{n/d}^\infty$, where $c$ is the nilpotency class of $A$, $d$ is thelargest divisor of $n$ coprime with $m$, $C_{d}^c$ is the direct power of $c$copies of the cycle $C_d$ of order $d$, $C_{n/d}^\infty$ is the direct power ofcountably many copies of the cycle $C_{n/d}$ of order $n/d$. This criterioncontinues our previous work on cases when the similar criterions were given forwreath products of abelian groups or of finite groups. Also, this is ageneralization of known results in literature, which solve the same problem formuch more restricted cases. Some applications of the criterion are consideredat the end of paper.

A Classification Theorem for Varieties Generated by Wreath Products of Groups