Recall that for a pair $(X,A)$, there is the following long exact sequence
$\cdots \longrightarrow \pi_2 (A)\longrightarrow \pi_2 (X)\longrightarrow \pi_2 (X,A)\longrightarrow \pi_1 (A)\longrightarrow \pi_1 (X)\longrightarrow \pi_1 (X,A).$
Also, I know that $\pi_1 (A)$ acts on $\pi_n (X,A)$ and $\pi_n (X)$ for all $n$.
My question is that :
"is the above long exact sequence a sequence of $\mathbb{Z}[\pi_1 (A)]$-modules and $\mathbb{Z}[\pi_1 (A)]$-homomorphisms?
Could you help me to understand it?
Thank you in advance.
Yes. In fact, it is clear from the definitions that $\pi_1 (A)$ acts on the whole long exact sequence of homotopy groups for $(X,A)$, the action commuting with the various maps in the sequence. (see Hatcher's book, p. 345).