In this article, we de. ne a module M to be G-extending if and only if for each X <= M there exists a direct summand D of M such that X boolean AND D is essential in both X and D. We consider the decomposition theory for G-extending modules and give a characterization of the Abelian groups which are G-extending. In contrast to the characterization of extending Abelian groups, we obtain that all finitely generated Abelian groups are G-extending. We prove that a minimal cogenerator for Mod-R is G-extending, but not, in general, extending. It is also shown that if M is (G-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.