Quotient-balanced implies quotient-transitive

Statement
Suppose $$a$$ is a property of functions from a group to itself. Suppose $$p$$ is the quotient-balanced subgroup property corresponding to $$a$$. In other words, a subgroup $$H$$ of a group $$G$$ satisfies property $$p$$ in $$G$$ if $$H$$ is a normal subgroup of $$G$$ and any function from $$G$$ to itself satisfying property $$a$$ sends cosets of $$H$$ to cosets of $$H$$ and the induced map on $$G/H$$ also satisfies property $$a$$.

Then, $$p$$ is a quotient-transitive subgroup property: if $$H \le K \le G$$ are groups such that $$H$$ satisfies property $$p$$ in $$G$$ and $$K/H$$ satisfies property $$p$$ in $$G/H$$, then $$K$$ satisfies property $$p$$ in $$G$$.

Related facts

 * Balanced implies transitive

Particular cases

 * Normality is quotient-transitive
 * Characteristicity is quotient-transitive
 * Strict characteristicity is quotient-transitive
 * Full invariance is quotient-transitive