Group actions - modulo 4

Question

I am having a bit trouble understanding group actions. if I am given a set A = {a,b,c,d} and a group action s: Z mod 4 -> $S_A$, how would one then be able to show if there exists a group action s such that s(2) = (a b). The only thing I seem to be able to get out of this information is that: $s(f +_4 g) = s_f \circ s_g $ and $s(e) = e_A$. But I don't know if that is useful in any way.

Answer 1

Hint : Look at the image of $1$ mod $4$ through $s$.

The order of $s(1)$ may be $1$, $2$ or $4$. If the image $s(1)$ is of order $1$ or $2$ then $s(2)=s(1)^2$...

If the image $s(1)$ is of order $4$ then it is a $4$-cycle, $s(2)$ will be the square of a $4$-cycle... can it happen that this gives a transposition ?