The classical Wu formula claims that $$Sq^1(x_{d-1})=w_1(TM)\cup x_{d-1}$$ on a $d$-manifold $M$, where $x_{d-1}\in H^{d-1}(M,\mathbb{Z}_2)$.
I wonder whether there is a generalization of the classical Wu formula to general Bockstein homomorphisms. We consider the Bockstein homomorphism $$\beta_{(2,2^n)}:H^*(-,\mathbb{Z}_{2^n})\to H^{*+1}(-,\mathbb{Z}_2)$$ which is associated to the extension $\mathbb{Z}_2\to\mathbb{Z}_{2^{n+1}}\to\mathbb{Z}_{2^n}$.
I guess there is a generalized Wu formula: $$\boxed{\beta_{(2,2^n)}(x_{d-1})=\frac{1}{2^{n-1}}\tilde w_1(TM)\cup x_{d-1}}$$ on a $d$-manifold $M$, where $x_{d-1}\in H^{d-1}(M,\mathbb{Z}_{2^n})$.
Here $\tilde w_1(TM)$ is the twisted first Stiefel-Whitney class of the tangent bundle $TM$ of $M$ which is the pullback of $\tilde w_1$ under the classifying map $M\to BO(d)$. Let $\mathbb{Z}_{w_1}$ denote the orientation local system, the twisted first Stiefel-Whitney class $\tilde w_1\in H^1(BO(d),\mathbb{Z}_{w_1})$ is the pullback of the nonzero element of $H^1(BO(1),\mathbb{Z}_{w_1})=\mathbb{Z}_2$ under the determinant map $B\det:BO(d)\to BO(1)$.
The right hand side makes sense since $2\tilde w_1(TM)=0$.
Can you help me to prove or disprove the boxed formula above?
Thank you!
