Weak action representability of 2-nilpotent groups

In this article, we investigate the representability of actions of the category $\mathsf{Nil}_2(\mathsf{Grp})$ of $2$-nilpotent groups. We first provide an algebraic characterisation of derived actions in $\mathsf{Nil}_2(\mathsf{Grp})$ by determining a universal strict general actor of an object $X$, which turns out to be the group $\operatorname{Aut}_c(X)$ of central automorphisms of $X$. We also characterise the morphisms $B \to \operatorname{Aut}_c(X)$ that define an action of $B$ on $X$ in $\mathsf{Nil}_2(\mathsf{Grp})$. We then show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not action representable, and that the existence of a weak representation is related to the amalgamation property. Using the construction of an amalgam of a suitable family of abelian subgroups of $\operatorname{Aut}_c(X)$, we prove that the category $\mathsf{Nil}_2(\mathsf{Grp})$ is weakly action representable, and that a weak representing object can be chosen to be an abelian group. Finally, we show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not locally algebraically cartesian closed.