You might think that there cannot be an automorphism of ${\mathbb R}P^3$ that acts non-trivially on the degree one cohomology. But the bijection between the spin structures and degree one cohomology is not canonical, it's a torsor. The space ${\mathbb R}P^3$ is orientable because $3$ is odd, and an orientation reversing automorphism interchanges them. See Dabrowski and Trautman, "Spinor structures on spheres and projective spaces".