  We present a combinatorial proof for the existence of the sign refined grid
homology in lens spaces, and a self contained proof that
$\partial_{\mathbb{Z}}^2 = 0$. We also present a Sage program that computes
$\widehat{\mathrm{GH}} (L(p,q),K;\mathbb{Z})$, and provide empirical evidence
supporting the absence of torsion in these groups.
