As far as I can see, the group is not finite (except in some very trivial examples). Does this mean that I should also consider alternatives to Galois theory? Also, I am very much dependent on algorithmic solutions to this problem, since the map $F$ is quite complicated, that is numerators and denominators are polynomials of total degree $n-1$ or $n$ in the $n$ variables. Hence, it stands out of the question to deal with these maps without using a CAS like Maple or Mathematica.

alright, thanks for the clarification.

Thank you for your answer. I do however have no idea of Galois Theory (my traing has been mostly in integrable systems, mathematical physics, etc.) , so I do not even understand if - by your last two sentences - you mean whether this is a hard or easy problem...I guess I should start reading books on Galois theory :-)

Arbitrary $n$ would be great, but most of my examples are for $n=3,4,5,6$. The other involution does not act linearly. It is birational.