On involutions and generalized symmetric spaces of dicyclic groups

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

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