  Let $G=QD_{8k}~$ be the quasi-dihedral group of order $8n$ and $\theta$ be an
automorphism of $QD_{8k}$ of finite order. The fixed-point set $H$ of $\theta$
is defined as $H_{\theta}=G^{\theta}=\{x\in G \mid \theta(x)=x\}$ and
generalized symmetric space $Q$ of $\theta$ given by $Q_{\theta}=\{g\in G \mid
g=x\theta(x)^{-1}~\mbox{for some}~x\in G\}.$
  The characteristics of the sets $H$ and $Q$ have been calculated. It is shown
that for any $H$ and $Q,~~H.Q\neq QD_{8k}.$ the $H$-orbits on $Q$ are obtained
under different conditions. Moreover, the formula to find the order of $v$-th
root of unity in $\mathbb{Z}_{2k}$ for $QD_{8k}$ has been calculated. The
criteria to find the number of equivalence classes denoted by $C_{4k}$ of the
involution automorphism has also been constructed. Finally, the set of twisted
involutions $R=R_{\theta}=\{~x\in G~\mid~\theta(x)=x^{-1}\}$ has been explored.
