Policy gradient Reinforcement Learning (RL) algorithms have received much attention in seeking stochastic policies that maximize the average rewards. In addition, extensions based on the concept of the Natural Gradient (NG) show promising learning efficiency because these regard metrics for the task. Though there are two candidate metrics, Kakades Fisher Information Matrix (FIM) and Morimuras FIM, all RL algorithms with NG have followed the Kakades approach. In this paper, we describe a generalized Natural Gradient (gNG) by linearly interpolating the two FIMs and propose an efficient implementation for the gNG learning based on a theory of the estimating function, generalized Natural Actor-Critic (gNAC). The gNAC algorithm involves a near optimal auxiliary function to reduce the variance of the gNG estimates. Interestingly, the gNAC can be regarded as a natural extension of the current state-of-the-art NAC algorithm, as long as the interpolating parameter is appropriately selected. Numerical experiments showed that the proposed gNAC algorithm can estimate gNG efficiently and outperformed the NAC algorithm.