Graded components of local cohomology modules of invariant rings

Let $A$ be a regular domain containing a field $K$ of characteristic zero, $G$ be a finite subgroup of the group of automorphisms of $A$ and $B=A^G$ be the ring of invariants of $G$. Let $S=A[X_1,\dots ,X_m]$ and $R= B[X_1, \ldots, X_m]$bestandardgradedwith$\ deg \ A=0$,$\ deg \ B=0$and$\ deg
X_i=1$forall$i$\mathrm{.}Extendtheactionof$G$on$A$to$S$byfixing$X_i$. Note $S^G=R$. Let $I$ be an arbitrary homogeneous ideal in $R$. The main goal of this paper is to establish a comparative study of graded components of local cohomology modules $H_I^i(R)$ that would be analogs to those proven in a previous paper of the first author for $H_J^i(S)$ where $J$ is an arbitrary homogeneous ideal in $S$.