On involutions and generalized symmetric spaces of dicyclic groups

Let $G=\Dc_{n}$ be the dicyclic group of order $4n$. Let $\varphi$ be an automorphism of $G$ of order $k$. We describe $\varphi$ and the generalized symmetric space $Q$ of $G$ associated with $\varphi$. When $\varphi$ is an involution, we describe its fixed point group $H=G^{\varphi}$ along with the $H$-orbits and $G$-orbits of $Q$ corresponding to the action of $\varphi$-twisted conjugation.