topology of $S^{2}$-bundle with a section

Question

Let $B$ be an even dimensional closed manifold. Suppose that $\pi: E \rightarrow B$ is a fibre bundle with fibre $F = S^{2}$. Suppose furthermore that there is a section $s : B \rightarrow E$ (i.e. such that $\pi \circ s = id_{B}$). Is it true that $H_{2}(E,\mathbb{R})$ is generated by $H_{2}(s(B),\mathbb{R})$ and $H_{2}(F,\mathbb{R})$? (after the appropriate inclusions).