Preimage of Normal Subgroup of Quotient Group under Quotient Epimorphism is Normal

Theorem
Let $G$ be a group.

Let $H \lhd G$ where $\lhd$ denotes that $H$ is a normal subgroup of $G$.

Let $K \lhd G / H$.

Let $L = q_H^{-1} \sqbrk K$, where:
 * $q_H: G \to G / H$ is the quotient epimorphism from $G$ to the quotient group $G / H$
 * $q_H^{-1} \sqbrk K$ is the preimage of $K$ under $q_H$.

Then:
 * $L \lhd G$

and there exists a group isomorphism $\phi: \paren {G / H} / K \to G / L$ defined as:
 * $\phi \circ q_K \circ q_H = q_L$