  A $q$-analogue of a $t$-design is a set $S$ of subspaces (of dimension $k$)
of a finite vector space $V$ over a field of order $q$ such that each $t$
subspace is contained in a constant $\lambda$ number of elements of $S$. The
smallest nontrivial feasible parameters occur when $V$ has dimension $7$,
$t=2$, $q=2$, and $k=3$; which is the $q$-analogue of a $2$-$(7,3,1)$ design,
the Fano plane. The existence of the binary $q$-analogue of the Fano plane has
yet to be resolved, and it was shown by Kiermaier et al. (2016) that such a
configuration must have an automorphism group of order at most $2$. We show
that the binary $q$-analogue of the Fano plane has a trivial automorphism
group.
