Let G be a group and let $p$ be a prime number. If the set $mathcal{A}(G)$ of all commuting automorphisms of G forms a subgroup of Aut(G), then G is called $mathcal{A}(G)$-group. In this paper we show that any $p$-group with cyclic maximal subgroup is an $mathcal{A}(G)$-group. We also find the structure of the group $mathcal{A}(G)$ and we show that $mathcal{A}(G)=Aut_c(G)$. Moreover, we prove that for any prime $p$ and all integers $n{geq}3$, there exists a non-abelian $mathcal{A}(G)$-group of order $p^n$ in which $mathcal{A}(G)=Aut_c(G)$. If $p$ > 2, then $mathcal{A}(G)={cong}mathbb{Z}_p{ imes}mathbb{Z}_{p^{n-2}}$ and if $p=2$, then $mathcal{A}(G)={cong}mathbb{Z}_2{ imes}mathbb{Z}_2{ imes}mathbb{Z}_{2^{n-3}}$ or $mathbb{Z}_2{ imes}mathbb{Z}_2$.