11.46 (1990)

a) Does there exist a finite 3-group $G$ of nilpotency class 3 with the property $[a, a^\varphi] = 1$ for all $a \in G$ and all endomorphisms $\varphi$ of $G$? (See A. Caranti, J. Algebra, 97, no. 1 (1985), 1–13.)
b) Does there exist a finite $p$-group of nilpotency class greater than 2, with $\text{Aut}\,G = \text{Aut}_c\, G \cdot \text{Inn}\,G$, where $\text{Aut}_c\, G$ is the group of central automorphisms of $G$?
c) Does there exist a 2-Engel finite $p$-group $G$ of nilpotency class greater than 2 such that $\text{Aut}\,G = \text{Aut}_c\, G \cdot \text{Inn}\,G$?