Thank you very much for this very interesting answer.

@Alejandro In particular Iam thinking to lifting homeomorphism $f$ to $F$ you point out to.Thanks again for your attention

@Alejandro Thank you very much for your very interesting comment. I think about its details and also I try to remedy the post. regarding the surjectivity we may add extra assumption "compact connected Lie group"

@MoisheKohan please read my previous comment

@BenjaminSteinberg I wonder if the measurability of each algebraic conjugqcy class or topological conjugacy class is an obvious question

@BenjaminSteinberg What would be an apprpriate analogues of Caley theorem? On the other jand assume that $H$ is a closed subgroup of $G$. Assume that $a,b\in H$ are equivalent as elements of H are they equivalent as elements of $G$? this question is obvious for the algebraic conjugacy but what about topological congugacy?

@MoisheKohan I would appreciate if you give comment on the following question:

this situation remind me some papers of F.Ghahrqmany about extension of cerain maps between spheres of Banach space to linear maps between Banach space. I vaguely remember th e details. I read them about 10 or 15 years ago

@BenjaminSteinberg In fact the order can a non natural number as in the case of circle $\theta$ can be count as an order of $e^{i\theta}$ in $S^1$

@MoisheKohan Thank you for your correction

@Callum the title is changed.

@Callum yes you are right that one side of the Frobenius theorem is obvious. But I think that the question of this post is not trivial. in fqct it would be a good ideq I change the title to "extening the Frobenious Lie algebra of a Foliation". I think that this title is more clear. the aim is to extend the Lie algebra of foliation to a bigger Lie algebra so is the space if F-uniform vector fields a Lie algebra?

@BenjaminSteinberg Thank you for your interesting comment. Since "order" plays a crucial role so your comment is a motivation to consider a concept of order for elements of a topological group. The order is not a number but is an equivalent class.