  Let $G$ be a group acting continuously on a space $X$ and let $X/G$ be its
orbit space. Determining the topological or cohomological type of the orbit
space $X/G$ is a classical problem in the theory of transformation groups. In
this paper, we consider this problem for cohomology lens spaces. Let $X$ be a
finitistic space having the mod 2 cohomology algebra of the lens space
$L_p^{2m-1}(q_1,...,q_m)$. Then we classify completely the possible mod 2
cohomology algebra of orbit spaces of arbitrary free involutions on $X$. We
also give examples of spaces realizing the possible cohomology algebras. In the
end, we give an application of our results to non-existence of
$\mathbb{Z}_2$-equivariant map $\mathbb{S}^n \to X$.
