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On the ambient space of a Lie group with a left invariant metric that isisometric and isomorphic to a semidirect product$\mathbb{R}^2\rtimes_A\mathbb{R}$, we consider a domain $\Omega\subseteq\mathbb{R}^2\rtimes_A\{0\}$ and vertical $\pi$-graphs over $\Omega$ and studythe partial differential equation a function $u:\Omega \rightarrow \mathbb{R}$must satisfy in order to have prescribed mean curvature $H$. Using techniquesfrom quasilinear elliptic equations we prove that if a $\pi-$graph hasnon-negative mean curvature, then it satisfy some uniform height estimates thatdepend on $\Omega$ and on a parameter $\alpha$, given a priori. When$\text{trace}(A) > 0$, these estimates imply that the oscillation of a minimalgraph assuming the same constant value $n$ along the boundary tends to zerowhen $n\rightarrow + \infty$ and goes to $+ \infty$ if $n\rightarrow - \infty$.Furthermore, we use some of the estimates, allied with techniques from Killinggraphs, to prove the existence of minimal $\pi-$graphs assuming the value $0$along a piecewise smooth curve $\gamma$ with endpoints $p_1,\,p_2$ and havingas boundary $\gamma \cup(\{p_1\}\times[0,\,+\infty))\cup(\{p_2\}\times[0,\,+\infty))$.

The mean curvature equation on semidirect products $\mathbb{R}^2\rtimes_A\mathbb{R}$: Height estimates and Scherk-like graphs