Ker of \$pi:G/Nto G/H\$ (natural projection)

Let $$G$$ be a group and $$N$$ and $$H$$ are normal subgroup of $$G$$ and $$N$$ is normal subgroup of $$H$$.
$$pi:G/Nto G/H$$ be natural projection, that is, $$xpmod{N}to xpmod{H}$$.

Then, I would like to formally prove $$kerpi＝H/N$$.

My attempt:
From fundamental theorem on homomorphisms,
$$(G/N)/kerpi cong G/H$$