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.