  For every nonconstant rational function $\phi\in\mathbb{Q}(x)$, the Galois
groups of the dynatomic polynomials of $\phi$ encode various properties of
$\phi$ that are of interest in the subject of arithmetic dynamics. We study
here the structure of these Galois groups as $\phi$ varies in a particular
one-parameter family of maps, namely the quadratic rational maps having a
critical point of period 2. In particular, we provide explicit descriptions of
the third and fourth dynatomic Galois groups for maps in this family.
