We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group \(\mathbb{H}^n\), whose prototype is the Dirichlet problem for the \(p\)-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is \(p=2\), we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent \(s\) goes to \(1\).