On the residual of a factorized group with widely supersoluble factors

Let $\mathbb{P}$ be the set of all primes. A subgroup $H$ of a group $G$ is called {\it $\mathbb{P}$-subnormal} in $G$, if either $H=G$, or there exists a chain of subgroups $H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb P, \ \forall i.$Agroup$G$iscalledwidelysupersoluble,$$\mathrm{w}$$-supersolubleforshort,ifeverySylowsubgroupof$G$is$\mathbb P$-subnormalin$G$\mathrm{.}Agroup$G=AB$with$\mathbb P$-subnormal$$\mathrm{w}$$-supersolublesubgroups$A$and$B$ is studied. The structure of its $\mathrm{w}$-supersoluble residual is obtained. In particular, it coincides with the nilpotent residual of the $\mathcal{A}$-residual of $G$. Here $\mathcal{A}$ is the formation of all groups with abelian Sylow subgroups. Besides, we obtain new sufficient conditions for the $\mathrm{w}$-supersolubility of such group $G$.