  We consider the inverse mean curvature flow (IMCF) in the Heisenberg group
$(\He^n, d_\varepsilon)$, where $d_\varepsilon$ is distance associated to
either $| \cdot |_\varepsilon$, $\varepsilon>0$, the natural family of
left-invariant Riemannian metrics, or with their sub-Riemannian counterparts
for $\varepsilon=0$. For $\Omega \subseteq \He^n$ an open set with smooth
boundary $\Sigma_0=\partial \Omega$ satisfying a uniform exterior gauge-ball
condition and bounded complement we show existence of a global weak IMCF of
generalized hypersurfaces $\{\Sigma^\varepsilon_s\}_{s \geq 0} \subseteq
\mathbb{H}^n$ which is proper. Here, both in the Riemannian and in the
sub-Riemannian setting, we adopt the weak formulation introduced by Huisken and
Ilmanen in \cite{HuiskenIlmanen}, following the approach in \cite{Moser} due to
Moser and based on the the link between IMCF and $p$-harmonic functions.
