On the global quotient structure of the space of twisted stable maps to a quotient stack

Let ${\mathcal{X}}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack ${\mathcal{K}} _{g,n}({\mathcal{X}},d)$ of twisted stable maps is a quotient stack and can be embedded into a smooth Deligne-Mumford stack. When $G$ is finite, we give a more precise construction of ${\mathcal{K}}_{g,n}( {\mathcal{X}},d)$ using Hilbert schemes and admissible $G$-covers.