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

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. We show some desired properties of this formalism, among which: a compatibility with tame vanishing cycles over a (strctly) henselian trait, a compatibility with the theory of tame vanishing cycles over $\mathbb{A}^1_{S}$, a compatibility with tensor product and with duality. In the last section, we prove that monodromy-invariant vanishing cycles, introduced by the second named author, are the homotopy fixed points with respect to a canonical continuous action of $\mu_{\infty}$ of tame vanishing cycles over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$.