Induced map between homology $H_n(S^n) \to H_n(\mathbb RP^n)$

Question

Let $q:S^n\to \mathbb RP^n$ be the map that glues the antipodal points of $S^n$, I have a feeling that the induced map between homology $H_n(S^n) \to H_n(\mathbb RP^n)$ is the multiplication-by-2 map but I don't know how to verify it. For simplicity lets assure the coefficient group is $\mathbb Z_2$so that both groups are $\mathbb Z_2$, but I think this (the map is the multiplication-by-2) is true in more general coefficients.