I am trying to solve a problem of group action that
What is number of distinct actions of the group $\mathbb{Z}$ on the set $\{1, 2, 3, 4\}$?
Here my understanding is, for the group action there will be a permutation representative such that
$$\phi: \mathbb{Z}\rightarrow S_4$$
And to know the number of distinct group action I need to know how many possible $\phi$s are there.
Now, I am confused, how I can determine the the number of $\phi$, or am I going in wrong direction?