On equality of derival and inner automorphisms of some p-groups

For a group G, D(G) denotes the group of all derival automorphisms of G. For a finite nilpotent group of class 2, it is shown that $ D(G)\cong Hom(G/\gamma _{2}(G),\gamma _{2}(G)) $. We prove that if G is a nilpotent group of class $ \ge 3 $ such that $ Z(G)\subseteq \gamma _{2}(G) $ and $ D(G/Z(G))=Inn(G/Z(G)) $, then $ D(G)=Inn(G) $ if and only if $ Autcent(G)=Z(Inn(G)) $. Finally, for an odd prime p, we classify all p-groups of order $ p^{n}, 1\le n\le 5 $, for which $ D(G)= Inn(G) $