Strict characteristicity is quotient-transitive

Statement
Suppose $$H \le K \le G$$ 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 $$G$$.

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 $$H \le K \le G$$ such that $$H$$ is strictly characteristic in $$G$$ and $$K/H$$ is strictly characteristic in $$G/H$$.

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