Let $\pi_n(\mathbb{R}P^n)$ be the $n$-th homotopy group of the $n$-dimensional projective space. Then by the long exact sequence of homotopy groups associated to the fibration $S^n\to \mathbb{R}P^n\to B\mathbb{Z}_2$, I obtain $\pi_n(\mathbb{R}P^n)=\mathbb{Z}$. Moreover, $\pi_1(\mathbb{R}P^n)=\mathbb{Z}_2$.
Question: How to prove that the action of $\pi_1(\mathbb{R}P^n)$ on $\pi_n(\mathbb{R}P^n)$ is trivial if $n$ is odd is the action is non-trivial if $n$ is even?