$\pi_1$ action on relative homotopy groups $\pi_n(X,A)$

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.