Summary: In [the authors, Int. J. Approx. Reason. 83, 265–279 (2017; Zbl 1404.03024)] a new property of fuzzy implication functions, called the invariance property with respect to powers of a continuous t-norm, was introduced and its application in approximate reasoning was studied. In the same paper, the novel family of power based implications was presented as a family of fuzzy implication functions that satisfy such invariance property. Unfortunately, this fact is not entirely true since as we will prove in this paper, there exist some power based implications generated from specific ordinal sum t-norms which do not fulfil the invariance property. Thus, a characterization of which continuous t-norms can be used to generate power based implications satisfying the power based invariance property is presented. Additionally, as an alternative solution, we introduce in this paper a slight modification of this property in such a way that all power based implications satisfy it.