On equality of central and class preserving automorphisms of finite p-groups

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