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Let $E$ be an elliptic curve with a $p$-torsion point $Q$. Let $X= E \times \mu_p \times \mathbb G_m$. Let $\sigma$ be the automorphism of $X$ that sends $(P,a,b)$ to $(P+Q, a,ab)$. Then $\sigma$ has order $p$ and no fixed points, so the induced map to the quotient $Y=X/\sigma$ is etale.