  We consider countably many three dimensional
$\mathtt{PSL}_2(\mathbb{F}_7)$-del Pezzo surface fibrations over
$\mathbb{P}^1$. Conjecturally they are all irrational except two families, one
of which is the product of a del Pezzo surface with $\mathbb{P}^1$. We show
that the other model is $\mathtt{PSL}_2(\mathbb{F}_7)$-equivariantly birational
to $\mathbb{P}^2\times\mathbb{P}^1$. Based on a result of Prokhorov, we show
that they are non-conjugate as subgroups of the Cremona group
$\mathtt{Cr}_3(\mathbb{C})$.
