I was working on this problem, and I'm not convinced of part c. Since $Z_n$ is the cyclic group of order $n$, $Z_n = \langle x \rangle,$ so every element of $Z_n$ is of the form $x^r$, for some $r$.
However, for any automorphism of $Z_n$ that maps $x^{r_1} \mapsto x^{s_1}$, and $x^{r_2} \mapsto x^{s_2}$. Then $(x^{r_1})^a = x^{ar_1} = x^{s_1}$, which implies that $ar_1 \equiv s_1 \pmod{n}$ and similarly for $r_2$, and so on. But, doesn't $a \equiv s_1 {r_1}^{-1} \pmod{n}$ uniquely determine $a \pmod{n}$?