Semilinear automorphisms of classical groups and quivers

For a classical group $G$ over a field $F$ together with a finite-order automorphism $theta$ that acts compatibly on $F$, we describe the fixed point subgroup of $theta$ on $G$ and the eigenspaces of $theta$ on the Lie algebra $mathfrak{g}$ in terms of cyclic quivers with involution. More precise classification is given when $mathfrak{g}$ is a loop Lie algebra, i.e., when $F=mathbb{C}((t))$.