Fix an algebraic closure $\bar{\mathbf{Q}}$ of $\mathbf{Q}$.
Let $B\subset \mathbf{P}^1_{\bar{\mathbf{Q}}}$ be a closed subscheme of finite cardinality.
Let $K$ be a number field such that $B$ can be defined over $K$. Let $B_K$ be a closed subscheme of $\mathbf{P}^1_K$ such that the base change to $\bar{\mathbf{Q}}$ is $B$.
Is the orbit of $B_K$ under the action of the absolute Galois group Gal$(\bar{\mathbf{Q}}/\mathbf{Q})$ finite? Is it a closed subscheme?
The answer is trivially yes if $K=\mathbf{Q}$. I expect the cardinality of the orbit to be less or equal to $[K:\mathbf{Q}]\cdot$#$B$ in general. But why?