What does normality is image closed mean here?

Question

So I $K,H$ are normal subgroups of G and K is a subgroup of H. Then H/K is normal in G/K.

The argument given was "because normality is image closed under quotient map by K normal subgroup of H and G gets sent to to a normal subgroup H/K of G/K.

I know that normality is image closed means that normal subgroups get sent to normal subgroups under homomorphism. But I don't understand how that is working here.

Answer 1

Consider the canonical homomorphism $h$ from $G$ to $G/K$. $h(H)=H/K\trianglelefteq G/K$, since $H\trianglelefteq G$.