Rational pullbacks of Galois covers

The finite subgroups of ${\rm PGL}_2(\mathbb{C})$ are shown to be the only finite groups $G$ with this property: for some integer $r_0$ (depending on $G$), all Galois covers $X\rightarrow \mathbb{P}^1_{\mathbb{C}}$ of group $G$ can be obtained by pulling back those with at most $r_0$ branch points along non-constant rational maps $\mathbb{P}^1_{\mathbb{C}} \rightarrow \mathbb{P}^1_{\mathbb{C}}$. For $G\subset {\rm PGL}_2(\mathbb{C})$, it is in fact enough to pull back one well-chosen cover with at most $3$ branch points. A consequence of the converse for inverse Galois theory is that, for $G\not \subset {\rm PGL}_2({\mathbb{C}})$, letting the branch point number grow provides truly new Galois realizations $F/{\mathbb{C}}(T)$ of $G$. Another application is that the ``Beckmann--Black'' property that ``any two Galois covers of $\mathbb{P}^1_{\mathbb{C}}$ with the same group $G$ are always pullbacks of another Galois cover of group $G$'' only holds if $G\subset {\rm PGL}_2({\mathbb{C}})$.

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