Quotient-transitive and stronger than normality implies complementary property is transitive

Statement

Suppose $p$ is a subgroup property stronger than the property of being a normal subgroup. Further, suppose $p$ is a quotient-transitive subgroup property: in other words, if $A\leq B\leq C$ are such that $A$ satisfies $p$ in $C$ and $B/A$ satisfies $p$ in $C/A$, then $B$ also satisfies $p$ in $C$.

Consider the subgroup property $q$ defined as follows: a subgroup $H$ of a group $G$ satisfies $q$ in $G$ if $H$ has a permutable complement in $G$ satisfying $p$ in $G$. Then, $q$ is a transitive subgroup property: if $H\leq K\leq G$ are groups such that $H$ satisfies $q$ in $K$ and $K$ satisfies $q$ in $G$, then $H$ also satisfies $q$ in $G$.