In 2013, Richard Pink studied properties of an inverse limit of Galois groups of iterated quadratic rational maps with infinite postcritical orbits whose critical points eventually meet. We use the theory of ramification groups to give an explicit representation of these Galois groups after an arbitrary number of iterations, and use this representation to describe the fixed point proportion of the Galois action on the roots of our functions. We also extend this work to certain rational functions of arbitrary degree.