Rational pullbacks of Galois covers

The finite subgroups of PGL ₂(C) are shown to be the only finite groups G with this property: for some integer r (depending on G), all Galois covers X→PC1 of group G can be obtained by pulling back those with at most r branch points along non-constant rational maps PC1→PC1. For G⊂ PGL ₂(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⊄ PGL ₂(C) , letting the branch point number grow provides truly new Galois realizations F/ C(T) of G. Another application is that the “Beckmann–Black” property that “any two Galois covers of PC1 with the same group G are always pullbacks of another Galois cover of group G” only holds if G⊂ PGL ₂(C).