Huisken and Ilmanen in  created the theory of weak solutions for inverse mean curvature flows (IMCF) of hypersurfaces on Riemannian manifolds, and proved successfully a Riemannian version of the Penrose inequality. In this paper we investigate and construct the sub-Riemannian version of the theory of weak solutions for inverse mean curvature flows of hypersurfaces in sub-Riemannian Heisenberg groups. We extend the weak solution theory in  to the first Heisenberg group and prove the existence, uniqueness and basic geometric properties of horizontal inverse mean curvature flows (HIMCF). By a Heisenberg dilation on HIMCF, we find a horizontal perimeter preserving flow (1.7) in the first Heisenberg group, and prove the existence and uniqueness of weak solutions to (1.7). Using this existential result, the present paper gives a positive answer to an open problem: Heintze-Karcher type inequality in the Heisenberg group. At the same time, this article also proves a Minkowski type formula in the first Heisenberg group.