  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
$[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$.
