We consider an inclusion $j: A \hookrightarrow X$. Let $A_j = A \times_X PX$ be the homotopy fiber (which is the fiber of the fibration associated to $j$). The space $PX$ there is the Moore path space $PX = \{ (\gamma,r) \in X^{[0,\infty)} \times (0,\infty): \gamma(t) = \gamma (r) ~ \forall t \geq r \}$ and $A \times_X PX = \{ (a,(\gamma,r)) \in A \times PX: \gamma (0) = a \}$ is the space of all paths starting in $A$.

The homotopy groups of $A_j$ are related to the relative homotopy groups of the pair $(X,A)$:
$$ \pi_n (X,A) \cong \pi_{n-1} (A_j) $$
(see Theorem 5.1.8 in
<cite authors="Piccinini, Renzo .">_Piccinini, Renzo A._, Lectures on homotopy theory, North-Holland Mathematics Studies. 171. Amsterdam etc.: North-Holland. xii, 293 p. (1992). [ZBL0742.55001](https://zbmath.org/?q=an:0742.55001).</cite>, for example).

My questions are: 

 1. Is there also an isomorphism $H_n(X,A) \cong H_{n-1} (A_j)$ on homology?
 2. If not, can one specify some conditions such that this statement holds?

Maybe one sentence about the background of my question: I want to find a Sullivan minimal model for the pair $(X,A)$ such that the rational Hurewicz morphism $\text{hur}_{(X,A)}: \pi_* (X,A) \otimes \mathbb{Q} \to H_* (X,A; \mathbb{Q})$ can be decoded via this model (compare to Chapter 13(c) of 
<cite authors="Felix, Yves; Halperin, Stephen; Thomas, Jean-Claude">_Felix, Yves; Halperin, Stephen; Thomas, Jean-Claude_, Rational homotopy theory, Graduate Texts in Mathematics. 205. New York, NY: Springer. xxxii, 535 p. (2001). [ZBL0961.55002](https://zbmath.org/?q=an:0961.55002).</cite>).

Thank you very much in advance.