Bijective transformations play an important role in generating fuzzy implications from fuzzy implications. In [Representations through a Monoid on the set of Fuzzy Implications, Fuzzy Sets and Systems, 247, 51-67], Vemuri and Jayaram proposed a monoid structure on the set of fuzzy implications, which is denoted by I, and using the largest subgroup S of this monoid discussed some group actions on the set I. In this context, they obtained a bijective transformation which ultimately led to hitherto unknown representations of the Yager's families of fuzzy implications, viz., f-, g-implications. This motivates us to consider whether the bijective transformations proposed by Baczyński & Drewniak and Jayaram & Mesiar, in different but purely analytic contexts, also possess any algebraic connotations. In this work, we show that these two bijective transformations can also be seen as being obtained from some group actions of S on I. Further, we consider the most general bijective transformation that generates fuzzy implications from fuzzy implications and show that it can also be obtained as a composition of group actions of S on I. Thus this work tries to position such bijective transformations from an algebraic perspective. © 2015 Elsevier B.V. All rights reserved.