Let $G$ be a Lie group acting smooth and effectively in a manifold $S$. Let $M$ be another manifold and $f:M \longrightarrow G$ a function such that the function: $$ F:M\times S \longrightarrow S$$ $$ F(x,s)=f(x)\cdot s$$ Is smooth.
Can we deduce that $f:M \longrightarrow G$ is smooth?
PS: for free actions or actions with a trivial isotropy group in some point $s\in S$ It's true. Because we have the diffeomorphism: $$ H: G \longrightarrow G\cdot s $$ $$ H(g)=g\cdot s$$ And recover locally $f$ by the smooth map: $$ x \mapsto H^{-1}\circ F(x,s)=f(x) $$
Is this argument generalizable to the general case?