I think I knew the answer to this once, but it's Friday afternoon here after a long week...

Let $X$ be a connected space. I want to study the stable Hurewicz homomorphism $$ h\colon\thinspace\pi_1^S(X) \to H_1(X;\mathbb{Z}). $$ I can see using the Atiyah-Hirzebruch spectral sequence that this is an epimorphism (I can also see this using the fact that the stabilisation homomorphism $\pi_1(X)\to \pi_1^S(X)$ factors through the abelianisation $\pi_1(X)\to H_1(X)$, which gives a splitting of $h$).

The spectral sequence also tells me that the kernel is either trivial or $\mathbb{Z}/2\mathbb{Z}=\pi_1^S({\rm pt})$.

I have vague memories of showing that the kernel is always $\mathbb{Z}/2\mathbb{Z}$, but I can't remember how. So I am half expecting an answer of "never" to the question

When is $h$ a monomorphism?