Finite factorizable groups with solvable $\mathbb P^2$-subnormal subgroups

A subgroup $H$ of a finite group $G$ is called $\mathbb P^2$-subnormal whenever there exists a subgroup chain $H=H_0\le H_1\le\dots\le H_n=G$ such that $|H_{i+1}:H_i|$ divides prime squares for all $i$. We study a finite group $G=AB$ on assuming that $A$ and $B$ are solvable subgroups and the indices of subgroups in the chains joining $A$ and $B$ with the group divide prime squares. In particular, we prove that a group of this type is solvable without using the classification of finite simple groups.