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We develop a theory of tame vanishing cycles for schemes over$[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$ in the context of \'etale sheaves. Weshow some desired properties of this formalism, among which: a compatibilitywith tame vanishing cycles over a (strctly) henselian trait, a compatibilitywith the theory of tame vanishing cycles over $\mathbb{A}^1_{S}$, acompatibility with tensor product and with duality. In the last section, weprove that monodromy-invariant vanishing cycles, introduced by the second namedauthor, are the homotopy fixed points with respect to a canonical continuousaction of $\mu_{\infty}$ of tame vanishing cycles over$[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$.

\'Etale tame vanishing cycles over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$