Let $X$ be a connected and locally simply connected topological space, and let $\widetilde{X}$ denote its universal cover. Assume additionally that the fundamental group of $X$ is finite, so that the covering $\pi:\widetilde{X}\to X$ is finite-sheeted. If $f:X\to X$ is a continuous involution (i.e. $f^2=\operatorname{id}_X$), then since $\widetilde{X}$ is simply connected, the map $f\circ\pi:\widetilde{X}\to X$ can be lifted to a map $\widetilde{f}:\widetilde{X}\to\widetilde{X}$ satisfying $f\circ\pi=\pi\circ\widetilde{f}$. Does this map $\widetilde{f}$ have a finite period, i.e. does there exist a positive integer $n$ such that $\widetilde{f}^n=\operatorname{id}_{\widetilde{X}}$?
I noted that $$\pi\circ\widetilde{f}^2=f\circ\pi\circ\widetilde{f}=f^2\circ\pi=\pi,$$ which means that $\widetilde{f}^2$ must preserve fibers. If I could show that it must actually permute each fiber, then we are done, since such a permutation must have finite order. But I couldn't seem show this on my own.
In fact, all what you need is that $\tilde{f}^2$ preserves fibers since it follows that there exists $\tilde{x}\in \tilde{X}$ and $n>0$ such that $\tilde{f}^{2n}(\tilde{x})=\tilde{x}$ (every self-map of a finite set has a periodic point). Now, the uniqueness property of the lifts implies that $\tilde{f}^{2n}=id$.
Incidentally, it is a general fact that homeomorphisms lift to homeomorphisms with respect to finite-to-one covering maps (not necessarily universal one), see this question and related this one, as well as this one for general universal covering maps. (There are probably more...)