Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally isomorphic to the stack quotient $[X/G]$ for $X$ a Hausdorff space and $G$ a finite group). Let's assume that $\mathfrak X$ is locally compact Hausdorff, i.e., for every etale map (or equivalently, one etale atlas) $X\to\mathfrak X$, where $X$ is a topological space, $X$ has an open cover by locally compact Hausdorff spaces.

**Is there a natural way to define the compactly supported cohomology groups $H^i_c(\mathfrak X,\mathbb Q)$?**

I would like the definition to agree with $H^i_c(X,\mathbb Q)^G$ for the simple case of a global quotient stack $\mathfrak X = [X/G]$, where $G$ is a finite group. Another property I'd like to have is a Thom isomorphism like statement, i.e., if we have an oriented rank $r$ vector bundle $\pi:\mathfrak E\to\mathfrak X$, then we have isomorphisms $\pi_!:H^{*+r}_c(\mathfrak E,\mathbb Q)\to H^*_c(\mathfrak X,\mathbb Q)$.