If $q$ is a fixed point, then clearly $q\in\Sigma$. Choose a tangent cone $Y_{q}=\mathbb{R}\times\mathbb{C}^{2}/\Gamma$ at $q$. The action of $\psi$ induces a one parameter group of isometric actions on $Y_{q}$, which fixes the origin. On the other hand on the smooth part of $Y_{q}$ the corresponding infinitesimal action is given by a Killing field of constant length. Clearly such a Killing field can not have zeroes. Contradiction. ∎