From page 6 of these solutions:

the differential\begin{equation}d_2: H_p(X,\Omega_1^{Spin})\rightarrow H_{p-2}(X,\Omega_2^{Spin})\end{equation}connecting the 1-st and the 2-nd row is the $\textbf{dual}$ of second Steenrod cohomology operation $Sq^2$.

Can someone please explain to me why this is true?