The primary objective of this study is to extend the concept of strongly I-deferred Cesàro summability and µ-deferred I-statistical convergence on amenable semigroups. Furthermore, under few conditions , we also establish some inclusion-based results. After that, we introduce the µ-deferred I *-statistical convergent, µ-deferred I-statistically pre-Cauchy, and µ-deferred I *-statistically pre-Cauchy functions in amenable semigroups and prove results based on connection among them. Certain counterexamples are also presented to support our results.