Assume $G$ is an Abelian group containing an element of order greater than $2$, then provide a nonidentity automorphism in $G$.
Let $\Psi:G \to G$ be a mapping given by $g \mapsto g^{\text{ord}(g_1)}$ and let $g_1 \in G$ be the element with order greater than $2$ then for $g,h \in G$ $$\Psi(gh)=(gh)^{\text{ord}(g_1)}=g^{\text{ord}(g_1)}h^{\text{ord}(g_1)}=\Psi(g)\Psi(h)$$
And this is indeed true since $G$ is Abelian.
Now we need to show that the kernel of the mapping contains only the identity element of $G$ denoted by $e$ but I think the mapping is not the one that is useful since $\Psi(g_1)=g_1^{\text{ord}(g_1)}=e$ which follows that $g_1 \in \text{ker}(\Psi)$ and $g_1$ is not necessarily identical to $e$ and hence the mapping is not necessarily an injection.
So what mapping will be useful and how can I finish the answer?
The mapping $\phi: g \mapsto g^{-1}$ is the desired automorphism. It is not equal to identity because $\phi(g_1)\not= g_1$.
Note that in the non-Abelian case, $\phi$ is not a morphism in $G$. It is a isomorphism from $G$ to its opposite group in this case.