Let \(\Gamma\) be a simple connected algebraic group over the field which is the algebraic closure of GF(p) (for some prime p). Let \(\sigma\) be an endomorphism of \(\Gamma\) and define \(G=\{g\in \Gamma:\) \(\sigma^ 2(g)=g\}\), \(S=\{g\in \Gamma:\) \(\sigma (g)=g\}\), and assume that G is finite. In this paper the authors investigate the action of G on the cosets of S, using the method suggested by R. Gow [Math. Z. 188, 45-54 (1984; Zbl 0564.20020)]. Their technique is then applied to two particular cases: the actions of \(B_ 2(q)\) on \((B_ 2(q):^ 2B_ 2(q))\) and of \(B_ 2(q^ 2)\) on \((B_ 2(q^ 2):B_ 2(q))\), where q is a power of 2. In the former, all suborbits are selfpaired. In the latter, there is precisely one pair of suborbits paired to each other; these become fused in the related action of Aut \(B_ 2(q)\). In each case, the rank, subdegrees and permutation character are given.