Finite Solvable Groups in Which the $\sigma$-Quasinormality of Subgroups is a Transitive Relation

Let $\sigma=\{\sigma_{i} \mid i\in I\}$ be a partition of the set of all primes, and let $G$ be a finite group. The group $G$ is said to be $\sigma$-primary if $G$ is a $\sigma_{i}$-group for some $i\in I$ and $\sigma$-complete if $G$ has a Hall $\sigma_{i}$-subgroup for each $i\in I$. A subgroup $A$ of $G$ is (i) $\sigma$-subnormal in $G$ if it has a subgroup series $A=A_{0} \leq A_{1} \leq \dotsb \leq A_{n}=G$ such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is ${\sigma}$-primary for each $i=1, \dots, n$; (ii) modular in $G$ if (1) $\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$ and (2) $\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $A \leq Z$; (iii) $\sigma$-quasinormal in $G$ if $A$ is $\sigma$-subnormal and modular in $G$. Finite solvable groups in which the $\sigma$-quasinormality of subgroups is a transitive relation are described. Some known results are generalized.