In a set of $n$ elements the number of bijective mappings is $n!$. In a group $G$ with $n$ elements, the number of automorphism is less than $n!$, because we require that any automorphism $f:G\to G$ to to satify $f(e) = e$, being $e$ the neutral element of the group. Thus we have that:
$$|\text{Aut}(G)| \le (n-1)!$$
However, almost certainly that upper bound is too high and can be improved. How can we give a better bound on the number of automorphisms of a general finite group?