Let $p:E \longrightarrow M$ be a smooth fibre bundle, with standard fibre space $F$ and $G$ a Lie group acting effectively on $F$ as a structure group.

Then, are the transition functions always smooth?

I mean, if for every open $U\subseteq M$ and function $g:U \longrightarrow G$ such that the function: $$ H:U\times F \longrightarrow U \times F$$ $$ H(x,s)=(x,f(x)\cdot s)$$ It's a diffeomorphism. Then, does it follow that $g$ is smooth?

PS: In Michor's book, natural operators in differential geometry, I read a similar fact in the step 5 of 9.11, but I don't see why it's true. https://www.mat.univie.ac.at/~michor/kmsbookh.pdf

Let $p:E \longrightarrow M$ be a smooth fibre bundle, with standard fibre space $F$ and $G$ a Lie group acting effectively on $F$ as a structure group.

Then, are the transition functions always smooth?

I mean, if for every open $U\subseteq M$ and function $g:U \longrightarrow G$ such that the function: $$ H:U\times F \longrightarrow U \times F$$ $$ H(x,s)=(x,g(x)\cdot s)$$ It's a diffeomorphism. Then, does it follow that $g$ is smooth?

PS: In Michor's book, natural operators in differential geometry, I read a similar fact in the step 5 of 9.11, but I don't see why it's true. https://www.mat.univie.ac.at/~michor/kmsbookh.pdf