e $$\begin{aligned} CC\left(I_{L}^{G}\mathcal{F}\right)&=CC(Ra_{\ast}(\widetilde{R\iota_{\ast}\mathcal{F}}))\\ &=a_{\ast}(CC(\widetilde{R\iota_{\ast}\mathcal{F}}))\\ &=a_{\ast}\left( \sum_{K_L-\mathrm{orbits }~S_L~\mathrm{in}~X_L}  \chi_{S_{L}}^{mic}(\mathcal{F})[\overline{T_{K\times_{K_L} \iota(S_{L})}^{\ast}(K\times_{K_L}X_G})]\right)\