This is problem of Rotman's Exercise 7.9(i).
If $G$ is finite abelian group with $|G| >2$, then $\operatorname{Aut}(G)$ has even order.
How can I approach to this problem? Could you suggest some hints?
Hint: By Lagrange's theorem, it suffices to find an element of $\operatorname{Aut}(G)$ with even order, or order $2$. Since $G$ is abelian, what do you know about inversion?