A note on Grid Homology in lens spaces: $\mathbb{Z}$ coefficients and computations

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