Birational geometry is the study of algebraic varieties up to a dense open subset. A classical question in this area is, given a variety X, to ask whether it is rational, i.e. birational to projective space. In recent decades, the importance of considering classical birational geometry questions in the presence of a group action has become very clear. Following this, a natural question arises: given a rational variety X of dimension n with the action of a (finite) group G, does there exist a G-action on P^n and a G-equivariant birational map X –> P^n? For n=2, this has been completely solved by Dolgachev-Iskovskikh and Pinardin-Sarikyan-Yasinsky, but for n > 2 this is still largely open. I will explain a resolution to the problem when X is a Fano threefold obtained by blowing up P^3 along a smooth genus 3 degree 6 non-hyperelliptic curve. This is joint work with Joe Malbon and Antoine Pinardin.