René Thom in 1952 proved the formula $$ Sq^i(U_2)=\Phi_2(w_i), $$ which in modern parlance says that the Steenrod squares of the mod $2$ Thom class of an orthogonal bundle are the images under the mod $2$ Thom isomorphism of the Stiefel-Whitney classes.

Apparently there is a mod $p$ analogue of this formula for oriented bundles, in which the $Sq^i$ are replaced by Steenrod powers $P^i_p$ and the $w_i$ are replaced by polynomials in the Pontryagin classes. Thom refers to this in his famous paper 1954 Commentarii paper; on page 49 he uses the instances $$ P^1_3(U_3)=\Phi_3(p_1),\qquad P^2_3(U_3)=\Phi_3(p_1^2+2p_2),\qquad P^1_5(U_5)=\Phi_5(p_1^2-2p_2), $$ and in a footnote at the bottom of page 49 claims that the calculations appear in the article

*Borel, Armand; Serre, Jean-Pierre*, **Groupes de Lie et puissances réduites de Steenrood**, Am. J. Math. 75, 409-448 (1953). ZBL0050.39603.

as well as some mimeographed notes bu Wu Wen Tsun with the title "Sur les puissances de Steenrod". Unfortunately I can't access Wu's notes, and the Borel-Serre paper only(!) gives the calculations of $P^i_p(p_k)$ from which I'm having trouble deducing the Thom class result. Hence my question:

**Question:** Does anyone know the general formula for $\Phi_p^{-1}P^i_p(U_p)$ in terms of the mod $p$ reductions of Pontryagin classes, where $U_p$ is the mod $p$ reduction of the Thom class for oriented bundles and $\Phi_p$ is the mod $p$ Thom isomorphism?