Transfer condition is composition-closed

Property-theoretic statement
The subgroup metaproperty called the transfer condition satisfies the subgroup metametaproperty of being composition-closed.

Statement with symbols
Suppose $$p,q$$ are two subgroup properties satisfying the transfer condition. Then, the composition $$p * q$$ also satisfies the transfer condition.

Transfer condition
A subgroup property $$p$$ is said to satisfy the transfer condition if whenever $$H \le G$$ has property $$p$$ in $$G$$, then for any subgroup $$K \le G$$, $$H \cap K$$ has property $$p$$ in $$K$$.

Composition operator
Given two subgroup properties $$p,q$$, the composition of $$p$$ and $$q$$, denoted $$p * q$$, is defined as follows. $$H \le G$$ has property $$p * q$$ in $$G$$ if there exists an intermediate subgroup $$K$$ of $$G$$ such that $$H$$ has property $$p$$ in $$K$$ and $$K$$ has property $$q$$ in $$G$$.