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A group $G$ in a variety $\mathfrak{V}$ is said to be absolutely-$\mathfrak{V}$, and we write $G\in A\mathfrak{V}$, if central extensions by $G$ are again in $\mathfrak{V}$. Absolutely-abelian groups have been classified by F. R. Beyl. In this paper we concentrate upon the class $A{\mathfrak{N}}_k$ of absolutely-nilpotent of class $k$ groups. We prove some closure properties of the class $A{\mathfrak{N}}_k$ and we show that every nilpotent of class $k$ group can be embedded in an $A{\mathfrak{N}}_k$-gvoup. We describe all metacyclic $A{\mathfrak{N}}_k$-groups and we characterize $2$-generator and infinite $3$-generator $A{\mathfrak{N}}_2$-groups. Finally...