A non-abelian simple group cannot act non-trivially on a set containing less than $5$ elements.

Question

My attempt: I understand my argument is wrong but I can't understand where am I making a mistake.

Let $G$ act on $S$ non - trivially.

$$\varphi:G \mapsto \mbox{Sym(S)}$$

$$\varphi(g) \mapsto \tau_g$$

Here $\tau_g:S \mapsto S$ is a bijective-map so that $\tau(g) = g.s$

We know that Ker($\varphi$) is a normal subgroup of $G$ that fixes all elements of $S$.

Then $[G:ker(\varphi)] > 1$ [$G$ acts non-trivially on $S$].

So there exists a non-trival normal subgroup of $G$ which is Ker($\varphi$).

I don't think the solution would be this simple.Where am I making a mistake?How do I rectify it?