Is it possible to put a supersymmetric theory on a lattice? Several lattice models have recently been shown to display emergent supersymmetry at length scales long enough that the lattice can be coarse-grained into a continuum (e.g. see here, here, and here).  Could a lattice model display exact supersymmetry even at the lattice length scales?  Clearly the answer is no, because the lattice breaks the Poincare subgroup of the supersymmetry group down to $S \times \mathbb{R}$, where $S$ is the lattice's space group and the $\mathbb{R}$ corresponds to time translational invariance.
According to this answer, the supersymmetry Lie supergroup $G$ corresponding to 3+1D SUSY with $N$ fermionic generators and no additional internal symmetries is Inonu-Wigner contracted $OSP(4/N)$.  Does there exist a Hamiltonian, defined on a lattice with space group $S$, with a symmetry Lie supergroup $H < G$ such that the bosonic part of $H$ is $S \times \mathbb{R}$ and the fermionic part of $H$ is nontrivial?  (In other words, I want to reduce the full supersymmetry supergroup $G$ down to a sub-supergroup $H$, such that the bosonic part of $G$ reduces from the Poincare group down to a lattice space group (times time translation), but without reducing the fermionic part all the way down to the identity, which would eliminate the supersymmetry entirely.)  This seems to me like the natural way to restrict supersymmetry to a lattice.
 A: " Could a lattice model display exact supersymmetry even at the lattice length scales? Clearly the answer is no." 
Yes, you are correct. In general, it is not possible. However, in some theories with extended supersymmetry, it is possible to maintain an exact nilpotent scalar supercharge exactly on the lattice. One can view this construction as follows - 1) take a continuum theory with sufficient number of supersymmetries, 2) twist the theory to get a topological field theory.
In flat space-time, this twisting is just a rewriting of the origin untwisted theory. Then, one can preserve a $\mathcal{Q}$, where $\mathcal{Q}^{2} = 0$. This prescription is valid for some theories, for ex: $\mathcal{N} = (2,2)$ SYM in two dimensions has now been studied on the lattice by various groups and in fact, in this theory one recovers all four supercharges as we take the continuum limit. In higher dimensions, this is still ongoing work. See this review : https://arxiv.org/abs/0903.4881
The ultimate goal is to study $\mathcal{N} = 4$ SYM in four dimensions. However, it seems that just by maintaining 1/16 of the supersymmetries on the lattice exactly, recovering other 15/16 is not an easy task. 
