Let $R$ be a regular local ring (I am particularly interested in the case when $R$ is the local ring of a point on a smooth scheme of finite type over a field). Let $G$ be the etale fundamental group of $Spec R$, i.e., the profinite group classifying the proper etale morphisms into $Spec R$.
Then to any finite abelian group with a $G$-module structure one can assign an etale sheaf on $Spec R$; this is a kind of inverse image functor. Conversely, to any etale sheaf of $\mathbb Z/m$-modules on $Spec R$ one can assign a $G$-module over $\mathbb Z/m$; this is a kind of direct image.
What can be said about these two functors? Is the direct image exact in this case? If I start with a $G$-module, take its inverse image to $Spec R$, and then take the direct image, do I get the same $G$-module that I started with?
Perhaps, a more concrete question: do the etale cohomology of $Spec R$ coincide with the profinite group cohomology of $G$? Say, with coefficients in a finite module over $G$, or even with finite constant coefficients?