  Given a simply connected solvable Lie group $G$, there always exists
NIL-affine action $\rho: G \to \operatorname{Aff}(H)$ on a nilpotent Lie group
$H$ such that $G$ acts simply transitively. The question whether this is always
possible for $H = \mathbb{R}^n$ abelian was known as Milnor's question, with a
negative answer due to a counterexample of Benoist. This counterexample is
based on a correspondence between certain affine actions $\rho: G \to
\operatorname{Aff}(\mathbb R^n)$ and left-symmetric structures on the
corresponding Lie algebra $\mathfrak g$ of $G$, where simply transitive actions
correspond exactly to the so-called complete left-symmetric structures. In
general however, the question remains open which solvable Lie groups $G$ can
act on which nilpotent Lie groups $H$.
  A natural candidate for a correspondence on the Lie algebra level is the
notion of post-Lie algebra structures, which form the natural generalization of
left-symmetric structures. In this paper, we show that every simply transitive
NIL-affine action of $G$ on a nilpotent Lie group $H$ indeed induces a post-Lie
algebra structure on the pair of Lie algebras $(\mathfrak g,\mathfrak h)$.
Moreover, we discuss a new notion of completeness for these structures in the
case that $\mathfrak h$ is $2$-step nilpotent, equivalent but different from
the known definition for $H = \mathbb R^n$. We then show that simply transitive
actions exactly correspond to complete post-Lie algebra structures in the
$2$-step nilpotent case. However, the questions how to define completeness in
higher nilpotency classes remains open, as we illustrate with an example in the
$3$-step nilpotent case.
