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Let $\mathcal{G}$ be a pro-algebraic group over $\mathbb{Q}_p$ with a continuous action of $G_K$ for a field $K$ (if $\mathcal{G}$ were an abelian unipotent group, this is precisely a $p$-adic Galois ...

This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples ...