On the deck groups of iterates of bicritical rational maps

Given a rational map $f:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ on the Riemann sphere, we define $\mathrm{D}\mathrm{e}\mathrm{c}\mathrm{k}(f)$ to be the group of Möbius transformations $\mu$ satisfying $f\circ \mu =f$. In this note, we consider the groups $\mathrm{D}\mathrm{e}\mathrm{c}\mathrm{k}(f^k)$, where $f$ is a \emph{bicritical} rational map (that is, a rational map with exactly two critical points) and $f^k$ denotes the $k$th iterate of $f$. In particular, we give a complete description of which groups (up to isomorphism) arise as the groups $\mathrm{D}\mathrm{e}\mathrm{c}\mathrm{k}(f^k)$ for bicritical rational maps $f$.