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.
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