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 $kerpiļ¼H/N$.

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