  We study post-Lie algebra structures on $(\mathfrak{g},\mathfrak{n})$ for
nilpotent Lie algebras. First we show that if $\mathfrak{g}$ is nilpotent such
that $H^0(\mathfrak{g},\mathfrak{n})=0$, then also $\mathfrak{n}$ must be
nilpotent, of bounded class. For post-Lie algebra structures $x\cdot y$ on
pairs of $2$-step nilpotent Lie algebras $(\mathfrak{g},\mathfrak{n})$ we give
necessary and sufficient conditions such that $x\circ y=\frac{1}{2}(x\cdot
y+y\cdot x)$ defines a CPA-structure on $\mathfrak{g}$, or on $\mathfrak{n}$.
As a corollary we obtain that every LR-structure on a Heisenberg Lie algebra of
dimension $n\ge 5$ is complete. Finally we classify all post-Lie algebra
structures on $(\mathfrak{g},\mathfrak{n})$ for $\mathfrak{g}\cong
\mathfrak{n}\cong \mathfrak{n}_3$, where $\mathfrak{n}_3$ is the
$3$-dimensional Heisenberg Lie algebra.
