Let $Spin(3)$ be embedded in $Spin(5)$ by the spin embedding then we have a fibration:
$Spin(3) \rightarrow Spin(5) \rightarrow Spin(5)/Spin(3)$
Where $Spin(5)/Spin(3)$ is $V_{5,2}$, the Stiefel manifold of orthonormal 2-frames in $\mathbb{R}^5$. Now we have $Spin(3)\cong S^3$ and $V_{5,3}$ is the sphere bundle in the tangent bundle of $S^4$ so the integral cohomology $H^*(V_{5,3})$ is $\mathbb{Z}$ at degree 0,7 and $\mathbb{Z}_2$ at degree 4.
Let $E$ be the Serre spectral sequence of the fibration, then $E^2_{4,3}=\mathbb{Z}_2$ but there can be no non-zero differentials to or from here. So $E^\infty_{4,3}\cong\mathbb{Z}_2$ and I get that $H^7(Spin(5))$ has some torsion. On the other hand $Spin(5)\cong Sp(2)$ and $H^*(Sp(2))\cong\Lambda(e_3,e_7)$ as it is a 3-sphere bundle over $S^7$ so $H^7(Spin(5))\cong \mathbb{Z}$
This is clearly a contradiction and I have messed up somewhere I would appriciate any help in finding my error.