Given a finite CW complex
X
X
, a nonzero cohomology class
ω
∈
H
1
(
X
,
Z
2
)
\omega \in H^1(X,\mathbb {Z}_2)
determines a double covering
X
ω
X^\omega
and a rank one
Z
\mathbb {Z}
-local system
L
ω
\mathcal {L}_\omega
. We investigate the relations between the homology groups
H
∗
(
X
ω
,
Z
)
H_*(X^{\omega },\mathbb {Z})
and
H
∗
(
X
,
L
ω
)
H_*(X,\mathcal {L}_\omega )
, when
X
X
is homotopy equivalent to a minimal CW complex. In particular, this settles a conjecture recently proposed by Ishibashi, Sugawara and Yoshinaga [Betti numbers and torsions in homology groups of double coverings, arxiv.org/abs/2209.02236, 2022, Conjecture 3.3], for a hyperplane arrangement complement.