Fix a space $X$, which I want to assume is a manifold. Under the assumption of simple-connectivity, Hurewicz's theorem tells us that $$ \pi_3(X)\to H_3(X,\mathbb{Z})\qquad (*) $$ is surjective, hence that every homology class is represented by a map $S^3\to X$. There is lots of hard work gone into worrying when homology classes are represented by embedded submanifolds, but that is not what I'm interested in here. What I want to know is:

Are there are nontrivial examples of manifolds $X$ with $(*)$ surjective, but with infinite $\pi_1(X)$?

I don't need the case when $\pi_1$ is finite, since I have a different proof that doesn't need this more subtle property.

An equivalent question is asking whether $H_3(\tilde{X},\mathbb{Z})\to H_3(X,\mathbb{Z})$ is surjective, for $\tilde{X}$ the universal covering space. Since $\pi_1(X)$ is nontrivial, then questions about the Eilenberg–Moore spectral sequence become more subtle, but I'm only looking at such a low-dimensional group that maybe things are not so bad (I don't really understand the EM spectral sequence, even relative to my background knowledge of spectral sequences, so I don't quite know how to start extracting information from that).

**Added:** I was interested to know if there are general hypothesis that allow me to conclude $(\ast)$ is onto, but given a comment below by user51223, I think all I care about is the weaker statement that
$$
\pi_3(X)\to H_3(X,\mathbb{Z}) \to H_3(X,\mathbb{Z})/\text{torsion} \qquad (**)
$$
is surjective. So:

Are there general conditions that guarantee this weaker statement is true?

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