I am studying SO(3) bundles on K3 surfaces, and in particular I want to understand several remarks made by Vafa and Witten in their article on S-duality in supersymmetric Yang-Mills theories. As is well known, SO(3) bundles are fully determined by their second Stiefel-Whitney class $w_2(E) \in H^2(K3,\mathbb{Z}_2)$ and the instanton number, which in this case can be written as $\frac14p_1$.

Now Vafa-Witten remark on page 55 of their article that K3 has a very large diffeomorphism group, which permute the possible values of $w_2$; they say that 'an obvious' diffeomorphism invariant of $w_2$ is $w_2^2\mod4$. But:

Why does $w_2$ change at all under a diffeomorphism? Any diffeomorphism $f$ from some space $X$ to K3 induces the pullback bundle $f^*E$ over $X$, but by the naturality of $w_2$, it would remain the same.

Then later on they say that modulo 2 the intersection form on $H^2(K3,\mathbb{Z})$ is equivalent to 11 copies of $H = \begin{pmatrix}0&1\\\\1&0\end{pmatrix}$. Any idea why this is the case? (The intersection form of K3 is of the form $2(-E_8)\oplus 3H$.)