  Let $G = H\times A$ be a group, where $H$ is a purely non-abelian subgroup of
$G$ and $A$ is a non-trivial abelian factor of $G$. Then, for $n \geq 2$, we
show that there exists an isomorphism $\phi : Aut_{Z(G)}^{\gamma_{n}(G)}(G)
\rightarrow Aut_{Z(H)}^{\gamma_{n}(H)}(H)$ such that
$\phi(Aut_{c}^{n-1}(G))=Aut_{c}^{n-1}(H)$. Also, for a finite non-abelian
$p$-group $G$ satisfying a certain natural hypothesis, we give some necessary
and sufficient conditions for $Autcent(G) = Aut_c^{n-1}(G)$. Furthermore, for a
finite non-abelian $p$-group $G$ we study the equality of $Autcent(G)$ with
$Aut_{Z(G)}^{\gamma_{n}(G)}(G)$.
