Let $G$ is non-abelian and simple group. Let $I ={\rm Inn}(G) \cong G$, $A = {\rm Aut}(G)$ and $B = {\rm Aut}(A)$. Since $Z(A)=1$, we have $A \cong {\rm Inn}(A)$, so we can identify $A$ with the normal subgroup ${\rm Inn}(A)$ of $B$.
Now $C:=C_B(I)$ is also a normal subgroup of $B$ and so $[C,A] \le C \cap A $.
Now my question is why $C \cap A=1$???
Again why $C:=C_B(I)$ is a normal subgroup of $B$??