most of the work here is showing that K/H is a subgroup of G/H. normality quickly follows.
(H is clearly normal in K, since it's normal in a larger group G).
so consider Hk, Hk' in K/H. is Hkk'^-1 is K/H?
what would it mean for K/H to be normal in G/H?
we would need (Hg)(K/H)(Hg)^-1 to be in K/H.
an element of (Hg)(K/H)(Hg)^-1 is the coset product: (Hg)(Hk)(Hg)^-1 = Hgkg^-1.
but K is normal in G, so....