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?