The exercise seeks to define for a fibration $F\to E\to B$ an action of $\pi_1(E)$ on $\pi_n(F)$, using the homotopy lifting property. However the basic idea imitating the action by deck transformations unsuprisingly leads to an action of $\pi_1(B)$ on $\pi_n(F)$.
I haven't found any ressources which explain how to construct the desired action, so any help would be appreciated, thanks!