$\pi_1$ 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.