  Let $R$ be a commutative Noetherian Henselian local ring. Denote by
$\mathrm{mod} R$ the category of finitely generated $R$-modules, and by
${\mathcal G}$ the full subcategory of $\mathrm{mod} R$ consisting of all
G-projective $R$-modules. In this paper, we consider when a given $R$-module
has a right ${\mathcal G}$-approximation. For this, we study the full
subcategory $\mathrm{rap}{\mathcal G}$ of $\mathrm{mod} R$ consisting of all
$R$-modules that admit right ${\mathcal G}$-approximations. We investigate the
structure of $\mathrm{rap}{\mathcal G}$ by observing ${\mathcal G}$, ${\mathcal
G}^{\bot}$ and $\mathrm{lap}{\mathcal G}$, where $\mathrm{lap}{\mathcal G}$
denotes the full subcategory of $\mathrm{mod} R$ consisting of all $R$-modules
that admit left ${\mathcal G}$-approximations. On the other hand, we also
characterize $\mathrm{rap}{\mathcal G}$ in terms of Tate cohomologies. We give
several sufficient conditions for ${\mathcal G}$ to be contravariantly finite
in $\mathrm{mod} R$.
