Implementation of a multigrid Montecarlo algorithm for a 2D Ising model

Question

In the paper "Monte Carlo Methods in Statistical Mechanics: Fundation and New Algorithms", the author A.D. Sokal explain how to merge the Montecarlo Markov Chain algorithmic approach with the multigrid one. Summarily the author explains firstly the multigrid algorithm in the deterministic framework, as a tool to solve $Au=f$ kind of problems using a Gauss-Siedel in the process. I'm adding a picture of the explantion of the algorithm in this deterministic case:

After that, the author states that it is possible to substitute the Gauss-Siedel relaxation step with an heath-bath algorithm to obtain this Montecarlo multigrid approach for statistical problems. This approach should be faster because the one-sites updates proper of the heath-bath algorithms act very slowly in presence of slow modes in the system, and the multigrid step should produce some collective updates on the system.

I want to write a Python code that performs a Montecarlo multigrid on a simple 2D Ising model, but I'm really struggling with the "coarsening" step. I really can't find a good way to coarse my Ising grid in such a way that, when the "uncoarsening" step of the muligrid occurs I don't have any ambiguity in the assignation of the spin values on the fine grid. And also is required for the values of the spins to be -1 or +1 also in the coarser grid.

And I am asking myself, is it in princible possible to apply this kind of algorithms to a simple 2D Ising model? Or maybe this algorithms are ok only for XY or gaussian models? Someone has some ideas on how to make this coarsening in a good way? I was thinking that I can change the spin-spin coupling constant somehow, but I really can't find a good way to do this without loosing information in the process.

Thanks a lot