Let $M$ be a path-connected space (say, a connected manifold) and denote by $\Omega_0 M$ the based loop space component of the constant loop $x_0$. If $c \colon M \to \Omega_0 M$ denotes the obvious map, then $c(M) \subseteq \Omega_0 M$ and one can speak of relative homotopy groups $\pi_k(\Omega_0 M,c(M))$ - to ease the notation we shall simply write $\pi_k(\Omega_0 M,M)$.
Now Hatcher describes (in great generality) how $\pi_1(A)$ acts on $\pi_k(X,A)$ for $A \subseteq X$ and goes on to prove some standard results. Let's call this the relative $\pi_1$-action.
I tried to understand this action in the above setting, i.e. $A=c(M)$ and $X=\Omega_0 M$, but fail to write down an explicit construction.
On the other hand, I tried to push the $\pi_1(M)$-action on $\pi_{k+1}(M)$ to $\pi_k(\Omega_0 M)$ using the usual isomorphism $$\pi_{k+1}(M) \cong \pi_k(\Omega_0 M).$$ (NB: this is not the connecting homomorphism from the usual loop space fibration!!) In this case I believe that the induced action becomes conjugation, i.e. $$[\gamma]\cdot [F]:=[\gamma \cdot F],$$ where $$\gamma \cdot F \colon S^k \to \Omega_0 M, \quad (\gamma \cdot F)(v)=\gamma^{-1} * F(v)*\gamma, \; v \in S^k.$$
In particular, this gives rise to a $\pi_1(M)$-action on $\pi_k(\Omega_0 M , M)$, begging the question:
Do the "relative" and "conjugation" actions of $\pi_1(M)$ on $\pi_k(\Omega_0 M,M)$ coincide?
I also tried to use the $H$-space structure of $\Omega_0 M$, but the latter does not seem to naturally enter the picture as we are acting via $\pi_1(M)$...