Strict characteristicity is quotient-transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., strictly characteristic subgroup) satisfying a subgroup metaproperty (i.e., quotient-transitive subgroup property)
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Statement

Suppose $H\leq K\leq G$ are groups such that $H$ is a strictly characteristic subgroup of $G$ and $K/H$ is a strictly characteristic subgroup of the quotient group $G/H$. Then, $K$ is a strictly characteristic subgroup of $G$.

Related facts

Generalizations

Quotient-balanced implies quotient-transitive. Other special cases of this are:

• Characteristicity is quotient-transitive
• Normality is quotient-transitive
• Full invariance is quotient-transitive

Proof

Given: Groups $H\leq K\leq G$ such that $H$ is strictly characteristic in $G$ and $K/H$ is strictly characteristic in $G/H$.

To prove: $K$ is strictly characteristic in $G$.