  Let $\Bbb P$ 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 $H=H_0\le H_1\le \ldots \le H_n=G, \ |H_{i}:H_{i-1}|\in \Bbb
P, \ \forall i.$ A group $G$ is called {\it widely supersoluble},
$\mathrm{w}$-supersoluble for short, if every Sylow subgroup of $G$ is $\mathbb
P$-subnormal in $G$. A group $G=AB$ with $\mathbb P$-subnormal
$\mathrm{w}$-supersoluble subgroups $A$ 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 $G$.
