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The map $\phi$ is in fact étale. If By definition a map $B \to [\mathcal X/G]$ (say $B$ is a map from a schemethen we get ) is a $G$-torsor $E \to B$ and a $G$-equivariant map $E \to \mathcal X$. The base change of $\phi$ is exactly $E \to B$. Since the property of schemes being étale is stable under base change and local, it is well known suffices to show that torsors for finite groups are étale (in fact $E\to B$ is étale. But part of the definition of being a torsor is being locally trivial in whatever topology you are considering, so now it is suffices to check that the map from $G$ to a trivial cover)point is étale, which is clear.

The general principle here is that properties of a group scheme $G$ should carry over to properties of the universal map $t \colon \mathrm{pt} \to BG$, e.g. if $G$ is smooth then $t$ is smooth.

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$\phi$ is in fact étale. If $B \to [\mathcal X/G]$ is a map from a scheme then we get a $G$-torsor $E \to B$ of schemes and it is well known that torsors for finite groups are étale (in fact étale locally it is a trivial cover).

The general principle here is that properties of a group scheme $G$ should carry over to properties of the universal map $t \colon \mathrm{pt} \to BG$, e.g. if $G$ is smooth then $t$ is smooth.