  Let $X$ be a connected finite CW complex. A connected double covering of $X$
is classified by a non-zero cohomology class $\omega \in H^1(X,\mathbb{Z}_2)$.
Denote the double covering space by $X^\omega$. There exists a corresponding
non-trivial rank one $\mathbb{Z}$-local system $\mathcal{L}_\omega$ on $X$.
What is the relation between the integral homology groups of $X^\omega$ and the
homology groups of the local system $\mathcal{L}_\omega$? When $X$ is homotopy
equivalent to a minimal CW complex, we give a complete answer to this question.
In particular, this settles a conjecture recently proposed by Ishibashi,
Sugawara and Yoshinaga for hyperplane arrangement complement. As an
application, when $X$ is a hyperplane arrangement complement and
$\mathcal{L}_\omega$ satisfies certain conditions, we show that
$H_*(X^\omega,\mathbb{Z})$ is combinatorially determined.
