My motivation, for reference: It's of interest to classify free actions of groups on spheres of positive even dimension. Establishing such a classification up to homotopy is quite straightforward: Every free group action on a sphere of even dimension is homotopic to either the trivial action by the trivial group or the antipodal action by $\mathbb{Z}/2$. The question in the title is equivalent to whether this classification is conserved if the "homotopic" in the above sentence is strengthened to "homeomorphic". This is by the following reduction:
Suppose that every space nontrivially double covered by $S^{2n}$ has the homeomorphism type of $\mathbb{P}^{2n}_{\mathbb{R}}$ and let $\tau\ \colon S^{2n}\to S^{2n}$ be some continuous involution lacking fixed points. Then:
By the compactness of $S^{2n}$, $$x\mapsto\text{dist}_{\text{standard subspace Euclidean metric on }S^{2n}}\left(x,\tau\left(x\right)\right)\colon S^{2n}\to\mathbb{R}_{\geq 0}$$ attains a nonzero mminimum on its domain.
By (1), the projection map $$\gamma\ \colon S^{2n}\to\text{coeq}\left(S^{2n}\substack{\overset{\text{id}}{\longrightarrow}\\ \underset{\tau}{\longrightarrow}}S^{2n}\right)$$ is a covering map.
By (2), there exists an isomorphism $$\psi\ \colon \text{coeq}\left(S^{2n}\substack{\overset{\text{id}}{\longrightarrow}\\ \underset{\tau}{\longrightarrow}}S^{2n}\right)\to\mathbb{P}_{\mathbb{R}}^{2n}.$$
By (2) and the lifting theorem for covering spaces, the $\psi$ of (3) lifts to an isomorphism $$\tilde{\psi}\ \colon S^{2n}\to S^{2n}$$ such that $$\lambda\circ\tau = \tau\circ\lambda,$$ precisely the desideratum.