Summary: “Given a map f: X → Y of compact Hausdorff spaces, the Mardeˇsić Factorization Theorem provides us a factorization f = qj, j: X → Z, q: Z → Y through a compact Hausdorff space Z with dim Z � dim X and weight of Z being at most weight of Y. The theorem has been generalized several times in various contexts with the Levin-Rubin-Schapiro Factorization Theorem being one of the most notable developments. “This paper introduces a new generalization in which the factoring space Z inherits the extension property for every map in the spirit of the Levin-Rubin-Schapiro Factorization Theorem. Such inheritance of extension properties is expressed by a new notion of extensional equivalence. Furthermore, we study the impact of such a generalization on the extension relation between maps, fτi.