Uniqueness of homomorphism from $S_A \rightarrow S_A$

Question

For any nonempty set $A$ the symmetric group $S_A$ acts on $A$ by $\rho a=\rho (a)$, for all $\rho \in S_A, a \in A$. The associated permutation representation (homomorphism from $S_A \rightarrow S_A$) is the identity map from $S_A$ to $S_A$.
Is this the only permutation representation? Why?

Answer 1

There are plenty of nontrivial automorphisms of $S_A$.

Each automorphism of $S_A$ will give a nontrivial permutation representation of $S_A$.