  We construct a $15$-dimensional affine variety $\Pi_{\mathbb{A}}^{15}$ with a
${\rm GL}_{2}$- and $(\mathbb{C}^{*})^{4}$-actions. We denote by
$\Pi_{\mathbb{A}}^{14}$ the affine variety obtained from
$\Pi_{\mathbb{A}}^{15}$ by setting one specified variable to $1$ (we refer the
precise definition to Definition 1.1 of the paper). Let $\Pi_{\mathbb{P}}^{13}$
be several weighted projectivizations of $\Pi_{\mathbb{A}}^{14}$, and
$\Pi_{\mathbb{P}}^{14}$ the weighted cone over $\Pi_{\mathbb{P}}^{13}$ with a
weight one coordinate added. We show that $\Pi_{\mathbb{P}}^{13}$ or
$\Pi_{\mathbb{P}}^{14}$ produce, as weighted complete intersections, examples
of prime $\mathbb{Q}$-Fano threefolds of codimension four belonging to the
eight classes No.308, 501, 512, 550, 577, 872, 878, and 1766 of the graded ring
database. The construction of $\Pi_{\mathbb{A}}^{15}$ is based on a certain
type of unprojection and is inspired by R.Taylor's thesis submitted to
University of Warwick. We also show that a partial projectivization of
$\Pi_{\mathbb{A}}^{15}$ has a $\mathbb{P}^{2}\times\mathbb{P}^{2}$-fibration
over the affine space $\mathbb{A}^{10}$. To show this, we introduce another
$13$-dimensional affine variety $H_{\mathbb{A}}^{13}$ whose product with an
open subset of $\mathbb{A}^{2}$ is isomorphic to a sextic cover of an open
subset of $\Pi_{\mathbb{A}}^{15}$.
