  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 $S= A[X_1,\ldots, X_m]$ and $R= B[X_1,
\ldots, X_m]$ be standard graded with $\ deg \ A=0$, $\ deg \ B=0$ and $\ deg \
X_i=1$ for all $i$. Extend the action of $G$ on $A$ to $S$ by fixing $X_i$.
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 $S$.

Commutative Algebra