Given $f:X \to Y$ a continuous map between two spaces (unpointed CW-complexes) such that $f$ induces an isomorphism in homology with integer coefficient, and $f$ induces an isomorphism on homology of the free loop spaces: $H_*(X^{S^1}) \to H_*(Y^{S^1})$ (also with integer coefficient). Does $f$ has to be a weak homotopy equivalence ?
If I further assume that $f$ induces an equivalence $H_*(X^{K}) \to H_*(Y^{K})$ for each finite CW-complex $K$, is $f$ a weak homotopy equivalence ?
Note that working in the unpointed settings makes a big difference here: It was showed by Arlin and Christensen that, contrary to the case of pointed connected space, for any small set of spaces $E$, there are maps that induces an isomorphism $\pi_0(X^K) \to \pi_0(Y^K)$ for all $K \in E$ without being weak equivalences. It is unclear to me if adding higher homology groups makes a big difference or not.
Edit: The case of groupoids $X=BG$ and $Y=BH$ is already an interesting example. $X^{S^1}$ is the (classyfing space of) the groupoid corresponding to the action of $G$ on itself by conjugation. Hence given $f:G \to H$ a morphism of group. Saying that it is a bijection on the $H_0$ of the loop space gives that $f$ induces a bijection between the set of conjugation classes of $G$ and $H$, and the fact that it is a bijection on the $H_1$ of the loop spaces gives that for all $g \in G$, $f$ induces an isomorphisms between the ablianization of the centralizer of $g$ and the abelianization of the centralizer of $h$.
That does not seem to be quite enough to conclude that $f$ is an isomorphisms, but this is already pretty restrictive. I havn't been able to formulate the fact that $f$ induce bijections on the higher $H_i$ in simple terms.
Now, if we start looking a $H_0(X^K)$ and $H_0(Y^K)$ for $K$ a finite complex, it seems that in this case all the interesting information is in the case where $K$ is a wedge of circle. Being a bijection on $H_0(X^K)$ and $H_0(Y^K)$ means that:
1) $f$ induces a bijection between $G^n/G$ and $H^n/H$ for all $n \in \mathbb{N}$, where $G$ and $H$ acts on $G^n$ and $H^n$ by the diagonal conjugation action.
2) For every finite family of elements $g_1,\dots,g_n$ in $G$, $f$ induce a bijection between the abelianization of the centralizer of $\{g_1,\dots,g_n\}$ and the abelianization of the centralizer of $\{f(g_1),\dots,f(g_n) \}$.
