If we consider arbitrary free involution on $S^n$, then the quotient need not be diffeomorphic to $\Bbb RP^n$ if $n\geq 5$ and a reference for this is "some curious involutions of spheres" by Morris W. Hirsch and John Milnor. In that same paper, it was mentioned as an unsolved problem that whether the quotient space is homeomorphic to $\Bbb RP^n$?. I would like to know if the answer of the above question is known by now and references for that will be very helpful.
Thank you so much!
In every dimension $\ge 4$ there is a fake real projective space, i.e., a manifold that is homotopy equivalent but not homeomorphic to $RP^n$.
Here you can find a computation for the topological surgery structure set of $RP^n$. It is stated for $n>4$ but I think it extends to $n=4$ because $\mathbb Z_2$ is good in the sense of Freedman. To get a homeomorphism classification of homotopy $RP^n$'s we also need to know the group of homotopy self equivalences of (any manifold homotopy equivalent to) $RP^n$: it is $\mathbb Z_2$ if $n$ is odd and trivial if $n$ is even (see e.g. p.61 of Rutter's survey ``Spaces of homotopy self-equivalences''). Moreover, if $n$ is odd, then $RP^n$ admits a self-map of degree $-1$, namely, the map induced by reflection in an equator of $S^n$. Thus any homotopy self-equivalence of $RP^n$ is homotopic to a diffeomorphism.
Let $f_i: M_i\to RP^n$ be homotopy equivalences representing different elements in the structure set. If $d_{ij}: M_i\to M_j$ is a homeomorphism, then $f_i^{-1}f_j d_{ij}$ is a homotopy self-equivalence of $M_i$. By the previous paragraph $f_i^{-1}f_j d_{ij}$ is homotopic to a homeomorphism, so $f_i$, $f_j$ represent the same element in the structure set. Thus in every dimension there is a fake $RP^n$.
A word of caution: the above argument does not show that the fake $RP^n$ is smoothable. For example, the only fake $RP^4$ is not smoothable as I learned from ``Invariant knots of free involutions of $S^4$'' by Ruberman, see here. There do exist many smoothable fake $RP^n$'s, see e.g., Smooth free involutions on homotopy $4k$-spheres by Fintushel-Stern.
