Find the number of group isomorphisms from the group $(\mathbb{Z}_3, +)$ to itself. I have to find the number of isomorphisms from the group $( \mathbb{Z}_3, + )$ to itself.
I don't know of any procedure to do this so I basically just tried to guess functions until I cannot find anything else. I found that the functions:
$$f(x) = x$$
$$f(x) = \hat{2}x$$
are both bijective and they hold the equality:
$$f(x + y) = f(x) + f(y)$$
true for any $x, y \in \mathbb{Z}_3$. So I concluded that these are all the isomorphism from the group $(\mathbb{Z}_3, +)$ to itself, since I couldn't find any more functions to be bijective and satisfy that condition.
I checked the answer of the exercise and it agrees with me, saying that the correct answer is $2$ (meaning $2$ isomorphisms, I guess the ones I found). 

My question is this: Is there a more organized, general way of finding the answer to this question.

Guessing all the possible function seems a bit weird, there are infinite possibilities.
How do I know that I found the maximum number of isomorphisms and that I can stop? Is there a better strategy than guessing the functions and coming up with the answer?

Hint: $${\rm Aut}(\Bbb Z_n)\cong U(n),$$ where $U(n)$ is the group of units modulo $n$.
