The Poincaré polynomial of an mp arrangement

Let $\mathcal{A}=\{A_i\}_{i\in I}$ be an mp arrangement in a complex algebraic variety $X$ with corresponding complement $Q(\mathcal{A})=X\backslash\bigcup_{i\in I}A_{i}$ and intersection poset $L(\mathcal{A})$. Examples of such arrangements are hyperplane arrangements and toral arrangements, i.e., collections of codimension 1 subtori, in an algebraic torus. Suppose a finite group $\Gamma$ acts on $X$ as a group of automorphisms and stabilizes the arrangement $\{A_i\}_{i\in I}$ setwise. We give a formula for the graded character of $\Gamma$ on the cohomology of $Q(\mathcal{A})$ in terms of the graded character of $\Gamma$ on the cohomology of certain subvarieties in $L(\mathcal{A})$.