A greedy block extended Kaczmarz method is introduced for solving the least squares problem where the greedy rule combines the maximum-distances with relaxation parameters. In order to save the computational cost of Moore-Penrose inverse, an average projection technique is used. The convergence theory of the greedy block extended Kaczmarz method is established and an upper bound for the convergence rate is also derived. Numerical experiments show that the proposed method is efficient and better than the randomized block extended Kaczmarz methods in terms of the number of iteration steps and computational time.