I am looking for a reference that addresses the question of which Hirzebruch surfaces $\mathbb{F}_n$ admit an involution that has only finitely many fixed points (no fixed curves).
Relatedly, I am also looking for information on the question of which blow-ups of Hirzebruch surfaces admit involutions with finitely many fixed points.
For some context, the question of which del Pezzo surfaces admit involutions with finite number of fixed points is addressed in Appendix A of this paper (it is only possible for $dP_3$ and $\mathbb{P}^1 \times \mathbb{P}^1$, the latter also being the Hirzebruch surface $\mathbb{F}_0$). I have not been able to find an analogous analysis for the Hirzebruch surfaces. In this paper on p.33 it is claimed without proof that such an involution can be found for $\mathbb{F}_2$, but that this cannot be done for all $\mathbb{F}_r$. Also the case of $\mathbb{F}_1$ is covered by the del Pezzo analysis, since $\mathbb{F}_1$ is isomorphic to $dP_1$.