What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?

What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?

I believe the answer for this is 6. As we can write the group elements as below

1. (a)(b)(C)
2. (ab)(c)
3. (ac)(b)
4. (bc)(a)
5. (abc)
6. (bac)

Can we generalize that for any $S_n$ onto $n$ the number of automorphisms will be $n!$ Also I cannot find this anywhere in my text, can we have a permutation $S_n$ onto $m$ where $n\neq m$? (for eg: is it possible to have a permutation group say $S_5$ on say 2 elements?)

What you did doesn't really work: you are listing elements of $S_3$, instead of automorphisms of $S_3$. While there is a natural isomorphism between the two, I suspect that if you are confusing elements and automorphisms, you are not expected to know this yet.

Try this:

1. If $f\colon S_3\to S_3$ is a group homomorphism, and you know what $f(12)$ and $f(123)$ are, then you know $f(\sigma)$ for all $\sigma\in S_3$.

2. If $f,g\colon S_3\to S_3$ are group homomorphisms and have the same values at $(12)$ and at $(123)$, then $f=g$.

3. $f(12)$ must be an element of order $2$. So there are, at most, three possibilities.

4. $f(123)$ must be an element of order $3$, so there are at most two possibilities.

5. Conclude that there are at most six automorphisms of $S_3$.

6. Exhibit six different automorphisms of $S_3$ (Hint: Conjugation...)