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 $K \lhd G / H$ and $L = q_H^{-1} \sqbrk K$, 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:
 * $L \lhd G$