Rational pullbacks of Galois covers

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