  In this paper, we investigate certain submodules in C*-algebras associated to
effective \'etale groupoids. First, we show that a submodule generated by
normalizers is a closure of the set of compactly supported continuous functions
on some open set. As a corollary, we show that discrete group coactions on
groupoid C*-algebras are induced by cocycles of \'etale groupoids if the fixed
point algebras contain C*-subalgebras of continuous functions vanishing at
infinity on the unit spaces. In the latter part, we prove the Galois
correspondence result for discrete group coactions on groupoid C*-algebras.
