Is there a Riemannian submersion from $Gl(2,\mathbb{R})$ to the Poincare half plane?

Let $ \mathbb{H}$ be the Poincare half plane with the hyperbolic metric. Let $ Gl(2,\mathbb{R})$ be equipped with a left invariant metric?

Is there a Riemannian submersion from $ Gl(2,\mathbb{R})$ to $ \mathbb{H}$ ? If yes, what is a precise formula for such a Riemannian submersion?