Let \(f(x)\) be a quadratic rational map defined over the field \(\mathbb{F}_q\). Then work of Pink (2013) and Juul, Kurlberg, Madhu, and Tucker (2015) classifies the possible Galois groups that arise from considering \(f^n(x)-t\) over the function field \(\mathbb{F}_q(t)\). For one class of Galois groups we describe the proportion of elements of with fixed points, and use a lesser known generalization of Burnside’s Lemma to show this is an upper bound across all classes. The Chebotarev Density Theorem translates this result to a bound on image set sizes.