Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Question

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows that $f$ induces nontrivial map of $H_n(\mathbb{R}P^n,\mathbb{Z}/2)$?

How do I see this best?