We consider the sparsity-promoting Linear Quadratic Gaussian (LQG) control for a network of linear systems. A sparse LQG controller requires fewer communication channels with the networked system. The bidirectional communication implies that the column-block- sparsity pattern of the Kalman filter and the row-block-sparsity pattern of the state feedback controller must be the same. We develop a two-step procedure: the first step being a sparsity- promoting design of the Kalman filter and the second step leveraging the structure identified in the first step to design the respective controllers.