  Let $A$ be a regular ring containing a field $K$ of characteristic zero and
let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $\deg A = 0$
and $\deg X_i = 1$ for all $i$. Let $G$ be a finite subgroup of $GL_m(A)$. Let
$G$ act linearly on $R$ fixing $A$. Let $S = R^G$. In this paper we present a
comprehensive study of graded components of local cohomology modules $H^i_I(S)$
where $I$ is an \emph{arbitrary} homogeneous ideal in $S$. We prove stronger
results when $G \subseteq GL_m(K)$. Some of our results are new even in the
case when $A$ is a field.
