Equivariant versal deformations of semistable curves

We prove that given any $n$-pointed prestable curve $C$ of genus $g$ with linearly reductive automorphism group ${\rm Aut}(C)$, there exists an ${\rm Aut}(C)$-equivariant miniversal deformation of $C$ over an affine variety $W$. In other words, we prove that the algebraic stack $\mathfrak{M}_{g,n}$ parametrizing $n$-pointed prestable curves of genus $g$ has an étale neighborhood of $[C]$ isomorphic to the quotient stack $[W / {\rm Aut}(C)]$.

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