In this work, we present a novel formulation for efficient estimation of group-sparse regression problems. By relaxing a covariance fitting criteria commonly used in array signal processing, we derive a generalization of the recent SPICE method for grouped variables. Such a formulation circumvents cumbersome model order estimation, while being inherently hyperparameter-free. We derive an implementation which iteratively decomposes into a series of convex optimization problems, each being solvable in closed-form. Furthermore, we show the connection between the proposed estimator and the class of LASSO-type estimators, where a dictionary-dependent regularization level is inherently set by the covariance fitting criteria. We also show how the proposed estimator may be used to form group-sparse estimates for sparse groups, as well as validating its robustness against coherency in the dictionary, i.e., the case of overlapping dictionary groups. Numerical results show preferable estimation performance, on par with a group-LASSO bestowed with oracle regularization, and well exceeding comparable greedy estimation methods.