The non-abelian tensor product of groups has its origins in the algebraic K-theory and homotopy theory. The nonabelian tensor product for a pair of groups is defined when the actions act compatibly on each other. This research is to determine the maximum number of a compatible pair of actions that can be identified between two cyclic groups of 2-power order for nonabelian tensor product. The compatible pair of actions between two cyclic groups of 2-power order can be found by using the necessary and sufficient conditions of two cyclic groups of 2-power order acting compatibly on each other. Hence, the number of the compatible pair of actions between two cyclic groups of the 2-power order is determined.