According to these solutions, the differential $d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})$ is the dual of $Sq^2$. Why?
This MO post asks a similar question (but about $d_3$ in K-theory), and the answer seems to be that there is a unique nonzero choice of stable cohomology operation. Is this also the case here? What about $d_2$ on other objects $E^2_{p,q}$? What about the differential $d_r$ on other pages?
The proof is given in the Lemma at p.751 of Teichner, "On the signature of four-manifolds with universal covering spin"MR1214960.