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It is known that there is a natural $\pi_1(X,x)$ action on $\pi_n(X,x)$ which also induces a bijection $\pi_n(X,x)/\pi_1(X,x) \cong [S^n, X]$.

Now, let $(X,A)$ be a pair of path-connected spaces and $x\in A$. We also have a $\pi_1(A,x)$ action on the relative homotopy group $\pi_n(X,A,x)$.

Is there any similar description for the quotient $\pi_n(X,A,x)/ \pi_1(A,x)$?

I guess it might be $[(D^n,S^{n-1}), (X,A)]$, the homotopy class of continuous maps of space pairs, since I notice that $\pi_n(X,A, x)$ can be defined as $[(D^n, S^{n-1},pt), (X,A,x)]$.

But maybe I was wrong, as I cannot find it in any reference.

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I'd say the only tricky part is showing the action is well-defined, but if you trust it is here is a proof following Hatcher's proof for the basepointed version:

Let's consider the general case of maps $(X,A,x_0) \rightarrow (Y,B,y_0)$ with $Y$ and $B$ path-connected. To be explicit the action is the following: given a loop $\gamma$ in $B$, it acts on $[f] \in \langle (X,A,x_0),(Y,B,y_0)\rangle$ by sending it to a class of a relative map $g$ that is homotopic to $f$ through a relative homotopy that moves the basepoint along $\gamma$.

We show that $\langle (X,A,x_0), (Y,B,y_0) \rangle / \pi_1(Y,y_0) \rightarrow [(X,A),(Y,B)]$ is a bijection:

It is surjective because given a relative map $f$, we can find a path from $f(x_0)$ to $y_0$ in $B$ which we can extend to a homotopy $A \times I \rightarrow B$ which we can extend to a homotopy $X \times I\rightarrow Y$.

It is injective by definition of the action since two classes that map to the same element are represented by maps that are homotopic through a relative homotopy taking a basepoint along a loop in $B$.

So taking $X=(D^n,S^{n-1})$ you get the result.

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