The question comes from the problem that the $H$ is a maximal normal subgroup of $G$ $\Leftrightarrow$ $G/H$ is simple. The overall proof is mostly done, but I feel some difficulty to understand the why does H contain in $K$ if f($K$)=$N$ and $N$ is a subgroup of the quotient group $G/H$. I am going to use this result to generate the "if" part use contradiction. I have already proved the converse part. It might be weird to stuck at such a strange place. Thanks for your help.
Since I have only learnt the 1-3 isomorphism theorem, I really don't have any idea about the fourth one. But I hope by the primary knowledge I could solve the problem.