Group zero-attracting LMS and its reweighted form have been proposed for addressing system identification problems with structural group sparsity in the parameters to estimate. Both algorithms however suffer from a trade-off between sparsity degree and estimation bias and, in addition, between convergence speed and steady-state performance like most adaptive filtering algorithms. It is therefore necessary to properly set their step size and regularization parameter. Based on a model of their transient behavior, we introduce a variable-parameter variant of both algorithms to address this issue. By minimizing their mean-square deviation at each time instant, we obtain closed-form expressions of the optimal step size and regularization parameter. Simulation results illustrate the effectiveness of the proposed algorithms.