Let $G$ be a finite group. The acentralizer of an automorphism $\alpha$ of $G$, is the subgroup of fixed points of $\alpha$, i.e., $C_G(\alpha)= \{g \in G \mid \alpha(g)=g\}$. In this paper we determine the acentralizers of the dihedral group of order $2n$, the dicyclic group of order $4n$ and the symmetric group on $n$ letters. As a result we see that if $n\geq 3$, then the number of acentralizers of the dihedral group and the dicyclic group of order $4n$ are equal. Also we determine the acentralizers of groups of orders $pq$ and $pqr$, where $p$, $q$ and $r$ are distinct primes.