Nonsmoothable involutions on spin 4-manifolds

Let $X$ be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to $n(-E_8)\bigoplus mH$, where $H$ is the hyperbolic form. In this paper, we prove that for $n$ such that $n\equiv 2 ~{\rm mod} ~4$, there exists a locally linear pseudofree $\mathbb{Z}_2$-action on $X$ which is nonsmoothable with respect to any possible smooth structure on $X$.