On the structure of Generalized Symmetric Spaces of $\bm{\mbox{SL}_{2}(\mathbb{F}_{\MakeLowercase{q}})}$} and $\bm{\mbox{GL}_{2}(\mathbb{F}_{\MakeLowercase{q}})}$

In this paper we will discuss the generalized symmetric spaces for $\SL_2(\mathbb{F}_q)$ and $\GL_2(\mathbb{F}_q)$. Specifically we will characterize the structure of these spaces and prove that when the characteristic of $\mathbb{F}_q$ is not equal to two the extended generalized symmetric space is equal to the generalized symmetric space for $\SL_2(\mathbb{F}_q)$ and nearly equal for $\GL_2(\mathbb{F}_q)$ for all but one involution.