On an open problem in the theory of modular subgroups

Let $G$ be a finite group. Then a subgroup $A$ of group $G$ is said to be modular in $G$ if $(i) \langle X,A\cap Z\rangle=\langle X,A\rangle\cap Z$ for all $X\leq G, Z\leq G$ such that $X\leq Z$, and $(ii)\langle A,Y\cap Z\rangle=\langle A,Y\rangle\cap Z$ for all $Y\leq G, Z\leq G$ such that $A\leq Z$. We obtain a description of finite groups in which modularity is a transitive relation, that is, if $A$ is a modular subgroup of $K$ and $K$ is a modular subgroup of $G$, then $A$ is a modular subgroup of $G$. The result obtained is a solution to one of the old problems in the theory of modular subgroups, which goes back to the works of A. Frigerio (1974), I. Zimmermann (1989).