Let $G$ be a finite group. Assume that $K \subseteq L \trianglelefteq G$ with $K \trianglelefteq G$. Then $L /K \subseteq \textbf{Z}(G / K)$ if, and only if $[G,L] \subseteq K$.
I know it is related to the three subgroup lemma, but I don't see the relation. Thanks in advance!