Quotient-balanced implies quotient-transitive

This article gives the statement and possibly, proof, of an implication relation between two subgroup metaproperties. That is, it states that every subgroup satisfying the first subgroup metaproperty (i.e., Quotient-balanced subgroup property (?)) must also satisfy the second subgroup metaproperty (i.e., Quotient-transitive subgroup property (?))
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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 $H\leq K\leq G$ 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