  By a result of Babai, with finitely many exceptions, every group $G$ admits a
semi-regular poset representation with three orbits, that is, a poset $P$ with
automorphism group $\textrm{Aut}(P) \simeq G$ such that the action of
$\textrm{Aut}(P)$ on the underlying set is free and with three orbits. Among
finite groups, only the trivial group and $\mathbb{Z}_2$ have a regular poset
representation (i.e. semi-regular with one orbit), however many infinite groups
admit such a representation. In this paper we study non-necessarily finite
groups which have a regular representation or a semi-regular representation
with two orbits. We prove that if $G$ admits a Cayley graph which is locally
the Cayley graph of a free group, then it has a semi-regular representation of
height 1 with two orbits. In this case we will see that any extension of the
integers by $G$ admits a regular representation. Applications are given to
finite simple groups, hyperbolic groups, random groups and indicable groups.
