Simple group with Klein four Sylow

Question

If $G$ is a simple group, with a Sylow $2$-subgroup isomorphic to the Klein four group $\mathbb{Z}_2 \times \mathbb{Z}_2$, then I want to show that any two involutions in a given Sylow $2$-subgroup are conjugate in $G$.

Any help would be appreciated.

Note: A solution/method in relatively elementary terms would be appreciated — I am alright with some machinery as long as you explain it.

Answer 1

I think you gate this from Sylow's 2nd theorem