Let $N$ be a normal subgroup of $G \times H$, and let $\pi_1: G \times H \to G$ and $\pi_2: G \times H \to H$ be the canonical projections. Then $\pi_1(N)$ is normal in $G$ and $\pi_2(N)$ is normal in $H$. What else can we say? I know that it is not true, in general, that $N \simeq \pi_1(N) \times \pi_2(N)$.

I'm particularly interested in the case where $G$ and $H$ are simple. In that case, $N \simeq \pi_1(N) \times \pi_2(N)$ except possibly in the case where $\pi_1(N) = G$ and $\pi_2(N) = H$. In that case, what do we know?