In this work, we perform an extension of the so-called Approximate Karush-Kuhn-Tucker (AKKT) condition, initially introduced in nonlinear programming [AHM11], for nonlinear symmetric cone programming. A new condition, which we call Trace AKKT (TAKKT), was also presented for the nonlinear semidefinite programming problem. TAKKT proved to be more practical than AKKT for nonlinear semidefinite programming. We prove that both the AKKT condition and the TAKKT condition are optimality conditions. Results of global convergence for the augmented Lagrangian method were obtained. Strict qualification conditions were introduced to measure the strength of the overall convergence results presented. Through these strict qualification conditions, it was possible to verify that our results of global convergence proved to be better than those known in the literature. We also present a proof for a particular case of the conjecture made in [AMS07].