Local cohomology modules of invariant rings

Let $K$ be a field and let $R$ be a regular domain containing $K$. Let $G$ be a finite subgroup of the group of automorphisms of $R$. We assume that $|G|$ is invertible in $K$. Let $R^G$ be the ring of invariants of $G$. Let $I$ be an ideal in $R^G$. Fix $i \geq 0$. If $R^G$ is Gorenstein then, \begin{enumerate} \item $injdim_{R^G} H^i_I(R^G) \leq \dim \ Supp \ H^i_I(R^G).$ \item $H^j_{\mathfrak{m}}(H^i_I(R^G))$ is injective, where $\mathfrak{m}$ is any maximal ideal of $R^G$. \item $\mu_j(P, H^i_I(R^G)) = \mu_j(P^\prime, H^i_{IR}(R))$ where $P^\prime$ is any prime in $R$ lying above $P$. \end{enumerate} We also prove that if $P$ is a prime ideal in $R^G$ with $R^G_P$ \textit{not Gorenstein} then either the bass numbers $\mu_j(P, H^i_I(R^G))$ is zero for all $j$ or there exists $c$ such that $\mu_j(P, H^i_I(R^G)) = 0$ for $j < c$ and $\mu_j(P, H^i_I(R^G)) > 0$ for all $j \geq c$.