On $K_p$-series and varieties generated by wreath products of $p$-groups

Let $A$ be a nilpotent $p$-group of finite exponent, and $B$ be an abelian $p$-groups of finite exponent. Then the wreath product $A\mathrm{W}\mathrm{r}B$ generates the variety $\mathrm{v}\mathrm{a}\mathrm{r}(A)\mathrm{v}\mathrm{a}\mathrm{r}(B)$ if and only if the group $B$ contains a subgroup isomorphic to the direct product $C_{p^v}^{\mathrm{\infty }}$ of at least countably many copies of the cyclic group $C_{p^v}$ of order $p^v=\exp (B)$. The obtained theorem continues our previous study of cases when ${\rm var}(A {\rm Wr} B ) = {\rm var}(A){\rm var}(B)$holdsforsomeotherclassesofgroups$A$and$B$ (abelian groups, finite groups, etc.).