  We examine the proposal in the previous paper to resolve the puzzle in
transitions in discrete gauge groups. We focus on a four-section geometry to
test the proposal. We observed that a discrete $\mathbb{Z}_2$ gauge group
enlarges and $U(1)$ also forms in F-theory along any bisection geometries locus
in the four-section geometry built as the complete intersections of two
quadrics in $\mathbb{P}^3$ fibered over any base. Furthermore, we demonstrate
that giving vacuum expectation values to hypermultiplets breaks the enlarged
$U(1)\times \mathbb{Z}_2$ gauge group down to a discrete $\mathbb{Z}_4$ gauge
group via Higgsing. We thus confirmed that the proposal in the previous paper
is consistent when a four-section splits into a pair of bisections in the
four-section geometry. This analysis may be useful for understanding the
Higgsing processes occurring in the transitions in discrete gauge groups in
six-dimensional F-theory models. We also discuss the construction of a family
of six-dimensional F-theory models in which $U(1)\times\mathbb{Z}_4$ forms.
