We can rewrite the problem as: Find the largest size of a set of integer-coordinate points such that, for any four points, their coordinate-wise sum does not have both coordinates divisible by 4. Firstly, we note that for the purposes of this problem we can treat two points as equal if both their coordinates are congruent $\bmod{4}$. This also means we can use a point multiple times, by adding some multiple of 4 to a coordinate. Then, if we take $3$ copies each of $(0, 0)$, $(0, 1)$, $(1, 0)$, $(1, 1)$, no four of them have a lattice point centroid. Next, we prove that no larger set can have this property. Suppose $13$ lattice points can be chosen. We classify them, according to the parity of both coordinates, into $4$ classes. For each class, we put its points into disjoint pairs; we can create at least $5$ pairs this way. The two points in each pair must sum to a point with both coordinates even. There are $5$ pairs and only $4$ unequivalent both-coordinates-even points, so some two pairs must sum to the same point with both coordinates even, and the sum of the four points has both coordinates divisible by 4.