Abstract: This talk is mainly about finite $p$-groups of nilpotency class $2$.Some structural information on such group is obtained in case when all central automorphisms of it are class-preserving. As a consequence some information is obtained about $p$-groups of class $2$ whose automorphisms are all class-preserving. I'll provide a brief history of the problem posed by A. Mann about finite $p$-groups whose automorphisms are all class-preserving, and pose few relevant problems and questions.