On involutions of type $\mathrm{O}(\mathrm{q},k)$ over a field of characteristic two

Symmetric spaces and their generalizations have played an important role in geometry, representation theory, $K$-theory, character sheaves, and other areas of mathematics over the last several decades. Let $G$ be a connected reductive group defined over $k$, a field of even characteristic, and let $H$ denote the fixed point group of $G$ with respect to an involutorial automorphism $\theta$. The symmetric $k$-varieties are the homogeneous spaces $G(k) / H(k)$, where $G(k)$ (respectively $H(k)$) denote the $k$-rational points of $G$ (resp. $H$). When $G$ is an orthogonal group defined over a field of characteristic 2, the involutions have been characterized when the corresponding quadratic space is nonsingular. We extend these results to defective and singular spaces. We discuss the conjugacy classes of these involutions, and conclude with a discussion of the fixed point groups of the involutions.