Let $(G,\ast )$ a group and $H$ a normal subgroup in $G$. If $K$ is a subgroup in $G$ containing $H$ then $K$ is normal in $G$

Question

I've tried show a counterexample using the dihedrical group of order 12, but the proposition seems true.

How I can start to prove that? My problem in the proof is show that the elements in $K-H$ satisfy that $x^-1 \ast k\ast x \in K ; \forall k\in K$.

I appreciate your help. Thank you.

Answer 1

Hint $H =\{ e \}$ is always normal in $G$.