$G$ is a group of finite order with center of odd cardinality. $G$ has a non-trivial subgroup $H$ that is simple and $[G:H]=2$. I want to prove that $H$ is the only non-trivial proper normal subgroup of $G$.
I think we should use this fact: if $N$ is normal in $G$ of cardinality 2 then $N
This is what I have done: $H$ is normal because of index 2. Take a subgroup $K$ normal, proper and non trivial. $K$ cannot be contained properly in $H$ because $H$ is simple. If I prove that $H\subseteq K$ then $|G/K||K/H|=|G/H|=2$ and so $K=H$. So I want to prove that $H\subseteq K$. Now, $H\cap K\trianglelefteq H$ and $H$ is simple then we have $H\cap K=\{e\}$ or $H\cap K=H$ and so $H\subseteq K$. So I suppose that $H\cap K=\{e\}$ and I want to reach a contradiction.
I don't know if this is the right approach, but I don't know how to continue.