  We classified prime $\mathbb{Q}$-Fano $3$-folds $X$ with only
$1/2(1,1,1)$-singularities and with $h^{0}(-K_{X})\geq 4$ a long time ago. The
classification was undertaken by blowing up each $X$ at one
$1/2(1,1,1)$-singularity and constructing a Sarkisov link. The purpose of this
paper is to reveal the geometries behind the Sarkisov links for $X$ in 5
classes. The main result asserts that any $X$ in the 5 classes can be embedded
as linear sections into bigger dimensional $\mathbb{Q}$-Fano varieties called
key varieties, where the key varieties are constructed by extending partially
the Sarkisov link in higher dimensions.
