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?
> 
> 
> 



Stable Hurewicz homomorphism in degree one
If by $\pi\_1^{st}X$ you mean the unreduced version (as you must when you say that $\pi\_1^{st}(pt)=\mathbb Z/2$), then for a based space $X$ you can functorially split off $\pi\_1^{st}(pt)$ from $\pi\_1^{st}X$, so there is always that kernel.


If you mean the reduced version, then you're precisely talking about the other factor in that splitting.


If we define $\pi\_k^{st}X$ as the direct limit of $\pi\_{k+n}(\Sigma^n X)$, then we get the reduced version. (Your $\mathbb Z/2$ may be considered as the unreduced $\pi\_1^{st}$ of a point, or as the reduced $\pi\_1^{st}$ of a $0$-sphere.)


As for the spectral sequence: You probably mean the one that goes from $H\_i(X;h\_j(X))$ to $h\_{i+j}(X)$ (everything unreduced) for any generalized homology theory. Here $X$ is not necessarily based or connected. If $X$ is connected then, choosing a basepoint, you can split off the left ($i=0$) column, so to speak. Or work with the analogous spectral sequence that goes from $H\_i(X,pt;h\_j(X))$ to $h\_{i+j}(X,pt)$ and lacks that column. 


That applies to any $h$. By looking separately at all components (or by choosing one point in each component in each component and then noting that this discrete set is a retract of $X$) you see that in general the canonical map $H\_0(X;\pi\_j(pt))\to h\_j(X)$ is a naturally split injection.


