Let S be the set of scalings {n(-1) : n = 1, 2, 3,...} and let L-z = zZ(2), z is an element of S, be the corresponding set of scaled lattices in R-2. In this paper averaging operators are defined for plaquette functions on L-z to plaquette functions on L-z' for all z', z is an element of S, z' = dz, d is an element of {2, 3, 4,...}, and their coherence is proved. This generalizes the averaging operators introduced by Balaban and Federbush. There are such coherent families of averaging operators for any dimension D = 1, 2, 3,... and not only for D = 2. Finally there are uniqueness theorems saying that in a sense, besides a form of straightforward averaging, the weights used are the only ones that give coherent families of averaging operators.772105123