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 $$A \le B \le C$$ 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 $$H \le K \le G$$ are groups such that $$H$$ satisfies $$q$$ in $$K$$ and $$K$$ satisfies $$q$$ in $$G$$, then $$H$$ also satisfies $$q$$ in $$G$$.