Normal subgroup of Normal subgroup If $H$ is a normal subgroup of $K$ and $K$ is a normal subgroup of $G$ can we say that $H$ is a normal subgroup of $G$.I could not prove it and cannot find a suitable counter example 

Will the results holds for $G$ abelian ?If else what will be counter example ?
It is not true that $H \lhd K \lhd G$ implies $H \lhd G$ (although the stronger condition that $ H \text{ char } K\ \lhd G$ will force $H \lhd G$).

A good tactic would be to choose a group $G$ with an abelian normal subgroup $K$. Any subgroup of $K$ must be abelian, hence normal in $K$. Your job is to find a subgroup of $K$ that's not normal in $G$.

The alternating group on $4$ letters, $A_4$ is a good choice for $G$. There, you can find such a chain $H \lhd K \lhd G$ with $H \not\lhd G$.
If $G$ is abelian then every subgroup its subgroup is normal: $gK = Kg$. Because $H$ also is subgroup of $G$ the answer is yes and proof is straightforward.