$O(n)$ is an extension of $\mathbb{Z}_2$ by $SO(n)$, $$1\to SO(n) \to O(n)\to \mathbb{Z}_2 \to 1.$$

Below we denote the Stiefel-Whitney class of real vector bundle $V_G$ of the group $G$ as: $$ w_j(V_{G}) : =w_j({G}). $$

My question is that how do the "generalized" Stiefel-Whitney class of $O(n)$ and $SO(n)$ relate to each other? What are related conversion formulas for $$ w_j(O(n)) = w_j(SO(n)) + ...? $$

What I have known are that:

1. $$ w_3({O(3)})=w_1({O(3)})^3+w_1({O(3)})w_2({SO(3)})+w_3({SO(3)}) $$ $$ {=w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_3(SO(3)) } $$ $$ =w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_3(SO(3)) $$ $$ {=w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_1(TM)w_2(O(3))} $$ $$ {=w_1(\mathbb{Z}_2)^3+w_1(\mathbb{Z}_2)w_2(SO(3))+w_1(TM)w_2(SO(3))} $$

2. $$ w_2({O(3)})=w_1({O(3)})^2+w_1({O(3)})w_2({SO(3)}) $$ $$ =w_1(\mathbb{Z}_2)^2+w_1(\mathbb{Z}_2)w_2({SO(3)}) $$

3.
When $n=1 \mod 4$, $$w_2(O(n)) = w_2(SO(n)) \mod 2, $$ When $n=3 \mod 4$,

$$ w_2(O(n)) = w_2(SO(n)) + w_1 \cup w_1 \mod 2, $$

The $w_1 \cup w_1$ is an obstruction to lifting $w_1$ to $\mathbb Z_4$ cohomology class.

Again, do we have

$$ w_j(O(n)) = w_j(SO(n)) + ...? $$

also, do we have

$$ w_2(O(2)) = w_2(SO(2))? $$ $$ w_j(O(2)) = w_j(SO(2))? $$

Edit for clarification: p.s. See eq. 2.5 of this paper, https://scipost.org/SciPostPhys.4.4.021/pdf -- I am using the same definition as theirs of “generalized” Stiefel-Whitney class of real vector bundles: $w_j(O(n))$ and $w_j(SO(n))$.