In the ambient space of a semidirect product
$\mathbb{R}^{2}\rtimes _{A}\mathbb{R}$
, we consider a connected domain
${\rm\Omega}\subseteq \mathbb{R}^{2}\rtimes _{A}\{0\}$
. Given a function
$u:{\rm\Omega}\rightarrow \mathbb{R}$
, its
${\it\pi}$
-graph is
$\text{graph}(u)=\{(x,y,u(x,y))\mid (x,y,0)\in {\rm\Omega}\}$
. In this paper we study the partial differential equation that
$u$
must satisfy so that
$\text{graph}(u)$
has prescribed mean curvature
$H$
. Using techniques from quasilinear elliptic equations we prove that if a
${\it\pi}$
-graph has a nonnegative mean curvature function, then it satisfies some uniform height estimates that depend on
${\rm\Omega}$
and on the supremum the function attains on the boundary of
${\rm\Omega}$
. When
$\text{trace}(A)>0$
, we prove that the oscillation of a minimal graph, assuming the same constant value
$n$
along the boundary, tends to zero when
$n\rightarrow +\infty$
and goes to
$+\infty$
if
$n\rightarrow -\infty$
. Furthermore, we use these estimates, allied with techniques from Killing graphs, to prove the existence of minimal
${\it\pi}$
-graphs assuming the value zero along a piecewise smooth curve
${\it\gamma}$
with endpoints
$p_{1},\,p_{2}$
and having as boundary
${\it\gamma}\cup (\{p_{1}\}\times [0,\,+\infty ))\cup (\{p_{2}\}\times [0,\,+\infty ))$
.