The paper studies spatial autoregressive models with group interaction structure, focussing on estimation and inference for the spatial autoregressive parameter λ. The quasi-maximum likelihood estimator for λ usually cannot be written in closed form, but using an exact result obtained earlier by the authors for its distribution function, we are able to provide a complete analysis of the properties of the estimator, and exact inference that can be based on it, in models that are balanced. This is presented first for the so-called pure model, with no regression component, but is also extended to some special cases of the more general model. We then study the much more difficult case of unbalanced models, giving analogues of some, but by no means all, of the results obtained for the balanced case earlier. In both balanced and unbalanced models, results obtained for the pure model generalize immediately to the model with group-specific regression components.