Let $L\colon G\times M\to M,(g,p)\mapsto gp$ be a smooth left group action of a smooth Lie group $G$ on a smooth manifold $M$. Trivially, $L_g:=L(g,\cdot)\colon M\to M$ is a diffeomorphism for every $g\in G$, and $L^p:=L(\cdot,p)\colon G\to M$ is smooth, but not necessarily bijective. We can fix the lack of bijectivity of $L^p$ by assuming $L$ to be simply transitive, that is, for any two $p,q\in M$ there is a unique $g_{pq}\in G$ such that $g_{pq}p=q$. My question now is whether $L^p$ is a diffeomorphism in this case?
Due to $g_{pq}p=q=L^p\bigl((L^p)^{-1}(q)\bigr)=(L^p)^{-1}(q)p$ for all $p,q\in M$ we have $(L^p)^{-1}=g_{p,(\cdot)}=\operatorname{inv}\circ\,g_{(\cdot),p}$ where the group inverseion $\operatorname{inv}\colon G\to G$ is a diffeomorphism which is its own inverse, so my question is equivalent to asking whether $g_{p,(\cdot)}\colon M\to G$ or $g_{(\cdot),p}\colon M\to G$ is smooth?