Let \(\mathbb{K}\) be a field of characteristic \(\neq2\), \(\varphi\in\mathbb{K}(x)\) be a quadratic rational function, and \(K_n\) be the splitting field of \(\varphi^n(x)-t\) over the function field \(\mathbb{K}(t)\). We consider the case where the two critical points \(c_1\) and \(c_2\) are wandering, yet there is some \(r\ge1\) such that \(\varphi^{r+1}(c_1)=\varphi^{r+1}(c_2)\). We prove that the elements of \(\operatorname{Gal}(K_{n+1}/K_{n})\) satisfy an “evenness” condition and use this to conclude that \(\operatorname{Gal}(K_{n+1}/K_0)\) is isomorphic to a semidirect product \(\operatorname{Gal}(K_{n+1}/K_{n})\rtimes\operatorname{Gal}(K_{n}/K_0)\). We use these results to bound the proportion of elements of \(\operatorname{Gal}(K_n/K_0)\) that fix a root of \(\varphi^n(x)-t\). If \(\mathbb{K}=\mathbb{Q}\), the Chebotarev Density Theorem for function fields translates these results to a bound on the image set size \(\#\varphi_p^n(\mathbb{P}^{1}(\mathbb{F}_p))\) where \(\varphi_p\) is the induced map on \(\mathbb{P}^{1}(\mathbb{F}_p)\). Similar results hold if \(\mathbb{K}\) is a number field.