It is natural. But I want to prove exactly.
My idea is to use isomorphism theorem.
Let $f: G \to G/H$ be the quotient map, then $f$ is a homomorphism and obviously surjective.
Hence, I just need to show $\ker(f) = K$.
However, I couldn't find the way. Definitely, $\ker(f) = H $ and $H \cong K$
But I can say $\ker (f) = K$.
How to prove it?