Degrees of maps factoring through real projective space

For $n\ge 2$, a self-map of $S^n$ factoring through Real projective space ${\mathbb{RP}}^n$ has degree zero when $n$ is even, and can have exactly the even degrees when $n$ is odd. In the odd case the first map lifts through the double covering and the covering has degree two. The dimension-one exception allows every integer, since ${\mathbb{RP}}^1\cong S^1$.