The Hurewicz map in motivic homotopy theory

For an ${\text{A}}^1$-connected pointed simplicial sheaf $\text{sX}$ over a perfect field $k$, we prove that the Hurewicz map ${\pi }_1^{{\text{A}}^1}(\text{sX})\to H_1^{{\text{A}}^1}(\text{sX})$ is surjective. We also observe that the Hurewicz map for ${\mathrm{\P }}_k^1$ is the abelianisation map. In the course of proving this result, we also show that for any morphism $\varphi$ of strongly ${\text{A}}^1$-invariant sheaves of groups, the image and kernel of $\varphi$ are also strongly ${\text{A}}^1$-invariant.