Faithfully flat quotient morphisms by $G_a$-actions on factorial affine varieties

Let $X$ be a factorial complex affine variety of dimension $\ge 3$ with an algebraic action of the additive group $G_a$. Let $\pi :X\to Y$ be the algebraic quotient morphism where we assume $Y$ is an affine variety. When $\pi$ is faithfully flat, we investigate $\pi$ by $G_a$-equivariant affine modifications and give criteria for $\pi$ to be a trivial ${\mathbb{A}}^1$-bundle. For a smooth acyclic fourfold $X$ with a free $G_a$-action and a $G_a$-equivariant ${\mathbb{A}}^3$-fibration $f:X\to {\mathbb{A}}^1$ where $G_a$ acts trivially on ${\mathbb{A}}^1$, we give a criterion for the algebraic quotient $Y$ to be isomorphic to ${\mathbb{A}}^3$ with $f$ as a coordinate. Together with a criterion for $\pi :X\to Y$ to be a trivial ${\mathbb{A}}^1$-bundle, we obtain a sufficient condition for $X\cong Y\times {\mathbb{A}}^1\cong {\mathbb{A}}^4$.