Cohomological behavior of the embedding $Gr(3,5)\to Gr(3,6)$

Let me give more details on Nanjun Yang's answer (which I think is correct). There is no need for any results concerning Chow-Witt rings of Grassmannians, the surjectivity can be deduced from the results of Casian and Kodama linked in Danny Ruberman's answer to this MO-question (or other sources of specific knowledge about the structure of the cohomology of the real Grassmannians).

Here's the relevant facts on the cohomology of the real Grassmannians ${\rm Gr}_k(\mathbb{R}^n)$. The non-torsion classes are generated by Pontryagin classes of the tautological sub- and quotient bundle, plus an additional class in degree $n-1$ in the case both $k$ and $n-k$ are odd (not a characteristic class but related to an Euler class on some other Grassmannian). All torsion is 2-torsion, and it's given by the image of the integral Bockstein from the mod 2 cohomology (i.e. generated by Bocksteins of Stiefel-Whitney classes). Relations between these generators all come in some way from the Whitney sum formula.

In the specific case we have ${\rm H}^4({\rm Gr}(3,5),\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$. A generator of the $\mathbb{Z}$-summand is the first Pontryagin class of either of the tautological bundles. A generator for the 2-torsion summand is the Bockstein of ${\rm w}_1^3$ (with ${\rm w}_1$ the first Stiefel-Whitney class of either of the tautological bundles). In Young diagram terms, this corresponds to the partition $(2,1,1)$. This can be explicitly found (for ${\rm Gr}(2,5)={\rm Gr}(3,5)$ in the paper of Casian and Kodama.

On the other hand we have ${\rm H}^4({\rm Gr}(3,6),\mathbb{Z})=\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}^{\oplus 2}$, generated again by the Pontryagin class and classes corresponding to the partitions $(3,1)$ and $(2,1,1)$. The restriction morphism induced from the inclusion ${\rm Gr}(3,5)\hookrightarrow{\rm Gr}(3,6)$ maps characteristic classes of the tautological bundles on ${\rm Gr}(3,6)$ to the corresponding characteristic classes of the tautological bundles on ${\rm Gr}(3,5)$. Because Pontryagin and Stiefel-Whitney classes are stable, we get the required surjectivity.

The general moral here is that we get surjectivity whenever the cohomology group of the smaller Grassmannian is generated by stable characteristic classes. The only problematic class is the non-characteristic class in degree $n-1$ (corresponding to the maximal hook partition). That only exists on ${\rm Gr}(k,n)$ whenever $k$ and $n-k$ are both odd. That is exactly precluded by the conditions in Nanjun Yang's answer.