I am interested in the low-degree stable homotopy group $\pi_2^{s}(X)$ of a path-connected space $X$. Using the Atiyah-Hirzebruch spectral sequence, we have the short exact sequence $0\to H_1(X,\mathbb{Z}/2)\to \pi_2^{s}(X)\to H_2(X,\mathbb{Z})\to 0$.

Question: Does it always split?

Edit: I've been working on a solution to my question, and I believe I've made progress. However, I've hit a roadblock in the proof and I'm hoping for some assistance. I will award the best answer to whoever can help me finish my proof or provide insight into my original question.

I think my exact sequence splits canonically! My idea is to show the following dual exact sequence splits canonically: $$ 0\to\mathrm{Hom}_{\mathbb{Z}}(H_2(X,\mathbb{Z}),\mathbb{Q}/\mathbb{Z})\to\mathrm{Hom}_{\mathbb{Z}}(\pi_2^s(X),\mathbb{Q}/\mathbb{Z})\stackrel{h}{\to} H^1(X,\mathbb{Z}/2)\to 0, $$ where I use the canonical isomorphism $\mathrm{Hom}_{\mathbb{Z}}(H_1(X,\mathbb{Z}/2),\mathbb{Q}/\mathbb{Z})\cong H^1(X,\mathbb{Z}/2)$. Next, I define $\phi$ as the generator of $\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(K(\mathbb{Z}/2,1)),\mathbb{Q}/\mathbb{Z})\simeq\mathbb{Z}/2$. For each $f\in H^1(X,\mathbb{Z}/2)\cong[X,K(\mathbb{Z}/2,1)]$, I consider the pullback $f^*\phi\in\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$. This defines a map $[\phi]:H^1(X,\mathbb{Z}/2)\mapsto\mathrm{Hom}_{\mathbb{Z}}(\pi^s_2(X),\mathbb{Q}/\mathbb{Z})$ that sends $f$ to $f^*\phi$.

I'm tempted to believe that $[\phi]$ is a homomorphism and $[\phi]\circ h=\mathrm{id}_{H^1(X,\mathbb{Z}/2)}$, but I cannot come up with a proof. If you have any insights into this, I would greatly appreciate it. Thank you!