Let $f$ a multi-valued function defined on the whole extended complex plane, which sends a finite set $A$ to a finite set $B$, that is, $f(A)=B$ and $f^{-1}(B)=A$.
Both $f$ and $f^{-1}$ may have only algebraic branch points or poles. For $f$, the set of its branch points and poles, if any, is a subset of $A$. For $f^{-1}$, the set of its branch points and poles, if any, is a subset of $B$. Everywhere else, $f$ and $f^{-1}$ are holomorphic.
Does there exist such a function $f$, if $A=\{0,1,4,\infty\}$ and $B=\{0,1,\infty\}$?
Note that the question is answered in positive for a function $g$, if $A=\{0,1,2,\infty\}$ and $B=\{0,1,\infty\}$, in which case $g(z)=1-(z-1)^2$.
Also note that the question may be always answered in positive if there is no requirement for the location of branch points. In such a case, the function $f$ can be implicitly defined by the equation $x(x-1)(x-4)=y(y-1)$, where $y=f(x)$.