For a finite group $G$, by the endomorphism ring of a module $M$ over a commutative ring $R$, we define a structure for $M$ to make it an $RG$-module so that we study the relations between the properties of $R$-modules and $RG$-modules. Mainly, we prove that $Rad_RM$ is an $RG$–submodule of $M$ if $M$ is an $RG$-module; also $Rad_RM\subseteq Rad_{RG}M$ where $Rad_AM$ is the intersection of the maximal $A$-submodule of module $M$ over a ring $A$. We also verify that $M$ is an injective (projective) $R$-module if and only if $M$ is an injective (projective) $RG$-module.