Group homomorphism between $\mathbb{Z}_2$ and ${\rm Aut}(G)$.

Question

Let $G$ be a group and ${\rm Aut}(G)$ its associated automorphism group. Prove that the function $$i:\mathbb{Z}_2\rightarrow{\rm Aut}(G)$$ given by $$i_0(h)=h,\;\;\;i_1(h)=h^{-1},$$ where $h \in G$, is a group homomorphism if and only if $G$ is abelian. I can't seem to figure out where to go with this question. If $i$ is a homomorphism, then $$i_0=i(0) = i(2) = i(1+1) = i(1)+i(1) =i_1\circ i_1$$ Similar equalities like this hold, but I don't see how they depend on $G$ being abelian. Any advice would be greatly appreciated.