$H = Z_5 $ and $G = S_5$ and let $A =\{1,2,3,4,5 \} .$
My question is
1.What is difference between the the action of H on A and G on A ?
2.How many group actions are possible in the above a cases ?
Let us take $a \in H ,$ then there is permutation $\sigma _ a$ acting on A. I can define $\sigma _a : A \rightarrow A $ by $\sigma_a(x) = a+x.$
My Doubt is that is this the only operation we can use for the action. I have used addition since $Z_5$ is a group under addition modulo 5.
On
An alternate way to regard action will allow you to see how many actions are possible: an action of a group $G$ on a set $A$ is the same as specifying a homomorphism of groups $\phi\colon G\to \mathrm{Perm\,}(A)$. where Perm(A) is the collection of all permutationa of the set A, regarded as a group wrt composition of functions.
Actions of $H$, say, on $A$ are described by homomorphisms $H \to S_{A} \cong S_{5}$.
In the first case, you can fix a generator $g$ of $H$, and the above homomorphism is described by the element $\gamma$ of $S_{5}$ to which you are mapping it. Now $\gamma$ must have order dividing $5$, so either it is the identity, or it has order $5$. Plently of choices, then. But you may want to consider actions up to similarities, and then the number drops quite drastically.
In the second case, you are considering homomorphisms $S_{5} \to S_{5}$. One is the trivial one, others have $A_{5}$ as its kernel (there are as many as there are involutions in $S_{5}$), and then there are the faithful ones (there are as many as there are automorphisms of $S_{5}$). Again, considering actions up to similarities is possibly the most sensible thing here.