To get you started, note that acts on its Sylow 2-subgroups. If there were 3 of them, then this would induce a homomorphism . On one hand, should not be trivial, because , which is too small. On the other, should not be , because the action is nontrivial. This means that is a proper nontrivial subgroup of . Since the kernel is always normal, this contradicts the simplicity of .
Try using a similar trick for part four.