Full invariance is quotient-transitive

Statement with symbols
Suppose $$H \le K \le G$$ are groups, such that $$H$$ is a fully invariant subgroup of $$G$$ and $$K/H$$ is a fully invariant subgroup of $$G/H$$. Then, $$K$$ is a fully invariant subgroup of $$G$$.

Generalization and other particular cases
A generalization of this fact is:


 * Quotient-balanced implies quotient-transitive

Other instances of this generalization are:


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

Other similar facts

 * Homomorph-containment is quotient-transitive
 * Subhomomorph-containment is quotient-transitive