  Let $G$ be a finite non-abelian $p$-group, where $p$ is a prime. Let
$\mathrm{Aut}_c(G)$ and $\mathrm{Aut}_z(G)$ respectively denote the group of
all class preserving and central automorphisms of $G$. We give a necessary
condition for $G$ such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G)$ and give
necessary and sufficient conditions for $G$ with elementary abelian or cyclic
center such that $\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G).$ We also characterize
all finite $p$-groups $G$ of order $\leq p^7$ such that
$\mathrm{Aut}_c(G)=\mathrm{Aut}_z(G)$ and complete the classification of all
finite $p$-groups of order $\le p^5$ for which there exist non-inner class
preserving automorphisms.
