Let $w_1\in H^1(M, \mathbb{Z}_2)$ be the Stiefel-Whitney class of the tangent bundle of d-dimensional manifold $M$, and $x_{d-1}\in H^{d-1}(M, \mathbb{Z}_2)$. Wu formula tells us $$Sq^1 x_{d-1}= u_1\cup x_{d-1}= w_1 \cup x_{d-1}$$ where $u_1$ is the first Wu-class.
Now, let $x_{d-1}\in H^{d-1}(M, \mathbb{Z}_N)$ for even $N$. Then we consider the Bockstein: $\beta: H^{d-1}(M, \mathbb{Z}_N)\to H^{d}(M, \mathbb{Z}_2)$. (When $N=2$, $Sq^1=\beta$. $\beta$ is a generalization of $Sq^1$ in the general $N$ case.)
The question is: whether there is a version of Wu formula for $$\beta x_{d-1}=?$$ for general even $N$?