Let $G$ be a Lie group and $\Gamma$ be a closed subgroup of $G$ and $\{g^t\}$ be a one-parameter subgroup of $G$.
Consider $\phi(\Gamma x, t):=\Gamma x g^t$. I wonder if $\phi:\Gamma \backslash G \times \mathbb R \to \Gamma \backslash G$ is automatically a smooth map (so that $\phi_t$ gives a smooth flow).
The key part is that whether $\{g^t\}$ has smooth properties immediately. Unfortunately I can't think of a counterexample.