  We construct several examples of genus-one fibered K3 surfaces without a
global section with type $I_{n}$ fibers, by considering double covers of a
special class of rational elliptic surfaces lacking a global section, known as
Halphen surfaces of index 2. The resulting K3 surfaces have bisection
geometries. F-theory compactifications on these K3 genus-one fibrations without
a section times a K3 yield models that have $SU(n)$ gauge symmetries with a
discrete $\mathbb{Z}_2$ symmetry.
