I have an excisive triad $(X;U,V)$ and consider $\delta^*:H^1(U\cap V)\to H^2(X)$ from the corresponding Mayer–Vietoris sequence. I'm wondering if there is a formula for the term $(\delta^*\alpha)^k\in H^{2k}(X)$.
For example, for the connecting homomorphism $\varepsilon^*:H^1(U)\to H^2(X,U)$ from the pair $(X,U)$, we know that it commutes mod 2 with the Steenrod squares, i. e. $(\varepsilon^*\alpha)^2=(\mathrm{Sq}^2\varepsilon^*)\alpha=\varepsilon^*\mathrm{Sq}^2\alpha$.
Maybe there is something similar for $\delta^*$?