For $n>2$, the outer automorphism group of $PSU(n)$ is $\mathbb{Z}_2$.
My question: Given a principal $PSU(n)$-bundle $P$ over a manifold $M$, can we extend the fiber of $P$ to $\mathbb{Z}_2\ltimes PSU(n)$? That is, can we find a principal $\mathbb{Z}_2\ltimes PSU(n)$ bundle over $M$? Here the semidirect product is given by the outer automorphism.
The case $n=2$ is also interesting, in this case, the question is: Given a principal $SO(3)$-bundle over a manifold $M$, can we find a principal $O(3)$-bundle over $M$?
For $n>2$, let $H=PSU(n)$, $G=\mathbb{Z}_2\ltimes PSU(n)$; For $n=2$, let $H=SO(3)$, $G=O(3)$.
Following Mike Miller's comment, we can construct a principal $G$-bundle $P\times_H G$ over $M$ from the principal $H$-bundle $P$ over $M$.
What is the relation between the Stiefel-Whitney classes of the associated vector bundles of $P\times_H G$ and $P$?
Thank you!