If $K\le H$ and $H \le G$ and $K\unlhd G$, is $H\unlhd G?$

Question

If $K\le H$ and $H \le G$ and $K\unlhd G$, is $H\unlhd G?$

Answer 1

The trivial group $1$ is a normal subgroup of every group $G$. Now just choose some group $G$ with some non-normal subgroup $1 \subsetneq H \subsetneq G$.

Answer 2

Not necessarily. Consider the most simple example: $K = \{ e \}$ and $H$ is any subgroup of $G$. Trivially, $K$ is a subgroup of $H$ and a normal subgroup of $G$, but $H$ might not be normal.