Let $F \to E \to B$ be a fibration. In Section 12 of Milnor and Stasheff, they consider a local system over $B$ $$ \{\pi_n(F)\} $$ whose fiber over $b \in B$ is the group $\pi_n(E_b)$.
I'm struggling to give a precise definition of this local system. It suffices to provide the data of a homomorphism $$ \pi_1(B, b_0) \to \operatorname{Aut}(\pi_n(E_{b_0}, e_0)), $$ or equivalently a homomorphism $$ \pi_1(B, b_0) \times \pi_n(E_{b_0}, e_0) \to \pi_n(E_{b_0}, e_0). $$ It seems the most reasonable way to define $\rho$ is as follows: given a loop $\gamma$ in $B$ and a sphere $f$ in $E_{b_0}$, use the homotopy lifting property to transport $f$ along the loop $\gamma$, the result of which is another sphere $\rho(\gamma, f)$ in $E_{b_0}$. But the problem is that the sphere $\rho(\gamma, f)$ may not be based at $e_0$ anymore, so this map isn't well-defined.
For $\rho$ to be well-defined, we need the loop $\gamma$ to admit a canonical (up to homotopy) lift to a closed loop in $E$. By the exact sequence of homotopy groups $$ \pi_1(F, e_0) \to \pi_1(E, e_0) \to \pi_1(B, b_0) \to \pi_0(F, e_0), $$ this is possible when the image of $\gamma$ in $\pi_0(F, e_0)$ is $0$ and we can choose a splitting of the surjection $\pi_1(E, e_0) \to \pi_1(B, b_0)$. But even if this splitting exists, it is not canonical unless there are non nontrivial homomorphisms $\pi_1(B, b_0) \to \pi_1(F, e_0)$ (e.g., when one of the two groups is trivial).
Question: Is there a general definition of the local system $\{\pi_n(F)\}$? If not, are the conditions above correct? I would be happy with a reference.
