What is the structure group of an associated bundle?

Question

Let $p:E\rightarrow B$ be a principal G-bundle such that G acts on a space F on the right. The "associated locally trivial bundle to E with fiber F" is the locally trivial fibration $$E\times_{G}F:=(E\times F)/G\rightarrow B$$ $$[y,f]\mapsto p(y)$$ with local trivialization induced by those for E. My question is: is the structure group of this new bundle the same of E? I would say that, since the associated bundle has fiber F its structure group should be just Diff(F) in the smooth category.In particular, if F=H<G is a subgroup of the Lie group G, structure group is H itself. Is it right? If not, can someone explain why?

Another question: if we start from a vector bundle $p:E\rightarrow B$ then we know that its structure group is $GL(n,\mathbb{R})$ and it's usually stated that one can build a principal bundle defining $$E_{GL}:=\bigcup U_{\alpha}\times GL(n,\mathbb{R})/\sim$$ where $\sim$ is the equivalence relation defined as $(b,f)\sim(\widetilde{b},\widetilde{f})\iff(\widetilde{b}=b,\widetilde{f}=f\phi_{\alpha\beta}(b))\forall b\in U_{\alpha}\cap U_{\beta}$, $\phi_{\alpha\beta}$ denoting the clutching functions of the initial vector bundle. My question is:- How can we be sure that this is a principal bundle? By definition the local trivializations should be equivariant etc.. and to me this is not obvious with that construction.

Answer 1

To answer the first question:
Yes, if $p:E \rightarrow B$ is a principal $G$-bundle and $G$ acts on $F$, the associated bundle $E \times_G F$ has same transition functions of $E$ (this follows from the fact that the local trivializations are induced by those of $E$), fiber $F$ and structure group $G$ (hence, the same of the principal $G$-bundle).
In particular, when we are talking about the reduction of the structure group to $H < G$, the bundle that we construct as $E' \times_H G$ has fiber $G$ and structure group $H$ (where we denote by $E'$ the principal $H$-bundle constructed via the transition functions with values in $H$).