  We will prove that given a genus-2 fibration $f: X \rightarrow C$ on a smooth
projective surface $X$ such that $b_1(X)=b_1(C)+2$, the fundamental group of
$X$ is almost isomorphic to $\pi_1(C) \times \pi_1(E)$, where $E$ is an
elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic
convexity of the universal cover of surfaces $X$ with genus-2 fibration
$X\rightarrow C$ such that $b_1(X)>b_1(C)$.
