Finite factored groups with soluble $\mathbb{X}$-subnormal factors

We establish new criteria of partial solubility of a finite factored group with restrictions on indices in chains of subgroups from factors to the group. In particular, it is proved that if $A$ and $B$ are soluble subgroups of a group $G$ such that there exist chains from $A$ and $B$ to $G$ in which indices of neighboring non-normal subgroups are either odd or equal to $2$ or $4$ and $G = AB$, then $G$ is soluble.