Affine connection defined by a quotient manifold?

Question

Suppose $G$ is a Lie group with affine connection $X,Y \mapsto\nabla_X Y\in C^{\infty}(G,TG)$, and $Q$ is a subgroup of $G$ such that $G/Q$ is also a nontrivial Lie group. Does this quotient manifold have a unique affine connection defined naturally by the affine connection on G? How do we construct it if so?