Let $Q$ be a simply-connected compact Lie group. Can one outline the proof (or provide the counter examples if my statement is false) that

there does not exist any group $G$ (with no topology) with a finite normal subgroup $N$, such that $G/N=Q$, where $N$ is not a direct factor of $G$?

Note that this is always false for $Q$ connected but not simply-connected.

(It is easy to come up with counter examples for connected Lie groups, such as $U(1)/(\mathbb{Z}/N\mathbb{Z})=U(1)$, $Spin(n)/(\mathbb{Z}/2\mathbb{Z})=SO(n)$. But those $Q$ are $U(1)$ and $SO(n)$, they are not simply-connected Lie groups.)

I would like to see that such finite group extensions cannot exist for $Q=\mathrm{SU}(N)$ explicitly. Perhaps more generally, one can argue that for $Q$ of type $A_n,B_n,C_n,D_n,E_6,E_7,E_8,F_4,G_2$, the above statement can be proved to be correct(?).