I want to prove the following proposition: Let $K$ be a normal subgroup of $G$ with $Z(K)=1$. Show that $G$ splits over $K$, if and only if $\operatorname{Aut}(K)$ splits over $\operatorname{Inn}(K)$.
I feel like I'm missing some tools to prove that. I know that $K/ Z(K)\simeq \operatorname{Inn}(K)$, thus $K\simeq \operatorname{Inn}(K)$.
I also know that $G/C_G(K)$ can be embedded in $\operatorname{Aut}(K)$ .
I thought that if I could show that every automorphism of $K$ is actually some conjugation by an element of $G$ then I could show immediately that if $G$ splits over $K$, then $\operatorname{Aut}(K)$ splits over $\operatorname{Inn}(K)$, but I'm not sure if it's actually true (that every automorphism of $K$ is actually some conjugation by an element of $G$).
I don't have any good idea about the inverse either.
Is there any key result relevant to this proposition that I haven't thought of using? Otherwise, could you give me some hint as to "where" to look for the complements I have to find?
Thank you for your help in advance!