I'm working my way through Mac Lane and Birkhoff Algebra and I have a question about one of their exercises. We are asked to show that certain automorphism groups are isomorphic (as groups) to a given group. For example, one part is to show that $\operatorname{Aut} (\mathbb{Z}_6) \cong \mathbb{Z}_2$. I'm fairly confident that I can write out all of the automorphisms and show that they are the only ones and show that they form a cyclic group of order (which we know from the reading is isomorphic to $\mathbb{Z}_2$). But, I'd like to argue this in a different way, that might save some time for the other parts of the problem. I'm curious if my solution is valid, or if I am missing something(s). Here it goes.
From a prior problem (coincidentally that I asked about a few days ago), we know that if $\phi: G \rightarrow H$ is an morphism between groups, then the image of $\phi$ forms a subgroup, $\operatorname{Im}(\phi) \subseteq H$. Next, since subgroups of cyclic groups are cyclic, we know that $\operatorname{Im}(\phi) = \langle a \rangle$, for some $a\in H$. Finally, if we are talking about automorphisms then, in particular, $\phi$ is an epimorphism. So, $\langle a \rangle =\operatorname{Im}(\phi) = H$. (Does this imply that generators of $G$ must be mapped to generator(s) of $H$?)
Thus, this problem reduces to finding the number of distinct generators of each group in question. For example, $\mathbb{Z}_6$ has two generators, $1$ and $5$. Hence, there are two distinct automorphisms, implying that the order of $\operatorname{Aut} (\mathbb{Z}_6) $ is two. The first is the identity automorphism, call it $\phi_{1}$ and the second, call it $\phi_2$ sends: $$ \begin{align} 1 &\mapsto 5\\ 2 &\mapsto 2\\ 3 &\mapsto 3\\ 4 &\mapsto 4\\ 5 &\mapsto 1\\ \end{align} $$ From this, $\phi_2 \circ \phi_2 = \phi_1$. So, $\operatorname{Aut} (\mathbb{Z}_6) = \left\{ \phi_1, \phi_2 \, | \, \phi_2^2 = \phi_1\right\} \cong \mathbb{Z}_2$. (Could we not get another automrphism by taking $\phi_2$ and instead of sending $2 \mapsto 2$ and $4 \mapsto 4$, send $2 \mapsto 4$ and $4 \mapsto 2$?)
As you can see, this solution/these ideas are not fully baked, so any assistance would be greatly appreciated. Thanks in advance.