Let be $A:=\{a,b,c,d\}$ and $\mathbb{Z}_4:=\{0,1,2,3\}$
Let s: $\mathbb{Z}_4 \rightarrow S_A$ a group action and $(\mathbb{Z}_4,+_4)$ where the operation is the addition modulo $4$.
1.Define $s_2:=(2)$ show that $s_2$ can be either the identity $e_A$ element or that the permutation $s_2$ has order $2$
Since $s$ is a group homomorphism: $$s(2+2)=s(2)\circ (2)$$ Since $2+2=0$ in $\mathbb{Z}_4$,$s$ map the identity element of $\mathbb{Z}_4$ to the idendity element in $S_A$ therefore $s(2)=s_2=e_4$ and it has order $2$ unless $s_2 = e_A$ and therefore it has order $1$
2.Assume $s_2 \ne e_A$ (identity element) and define s1. Show that $s_1$ has order $4$.
As above, we can see that $s(1+1+1+1)=s_1 \circ s_1 \circ s_1 \circ s_1= s(0)=e_A$ and therefore the order of $s_1$ is $4$.
3.Does exist a group action $s$: $\mathbb{Z}_4 \rightarrow S_A$ such that $s_2=(ab)$?
Suggestions for question 3?
The answer is no. The permutation $s_2$ is the square of the permutation $s_1$. So it must be an even permutation, but a single transposition is odd.