Quotient-balanced subgroup property

Definition
Suppose $$a$$ is a property of functions from a group to itself. Then, a normal subgroup $$H$$ of a group $$G$$ is said to satisfy the quotient-balanced subgroup property corresponding to $$a$$, if any map from $$G$$ to itself satisfying property $$a$$ sends every coset of $$H$$ to a coset of $$H$$, and the induced function from $$G/H$$ to itself also satisfies property $$a$$.