Let $f : [0,2] \rightarrow \mathbb{R}$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0) = 1$. Let
$$F(x) = \int_{0}^{x^2} f(\sqrt{t})\, dt$$
for $x \in [0,2]$. If $F'(x) = f'(x)$ for all $x \in (0,2)$, then $F(2)$ equals\\
(A) $e^2 - 1$\\
(B) $e^4 - 1$\\
(C) $e - 1$\\
(D) $e^4$