  Up to isomorphism, there is a unique connected semisimple algebraic group
over $\mathbb{Q}$ of Lie type $\mathrm{F}_{4}$, with compact real points and
split over $\mathbb{Q}_{p}$ for all primes $p$. Let $\mathbf{F}_{4}$ be such a
group. In this paper, we study the level one automorphic representations of
$\mathbf{F}_{4}$ in the spirit of the work of Chenevier, Renard, and Ta\"ibi.
First, we give an explicit formula for the number of these representations
having any given archimedean component. For this, we study the automorphism
group of the two definite exceptional Jordan algebras of rank $27$ over
$\mathbb{Z}$ studied by Gross, as well as the dimension of the invariants of
these groups in all irreducible representations of
$\mathbf{F}_{4}(\mathbb{R})$. Then, assuming standard conjectures by Arthur and
Langlands for $\mathbf{F}_{4}$, we refine this counting by studying the
contribution of the representations whose global Arthur parameter has any
possible image (or "Sato-Tate group"). This includes a detailed description of
all those images, as well as precise statements for the Arthur's multiplicity
formula in each case. As a consequence, we obtain a conjectural but explicit
formula for the number of algebraic, cuspidal, level one automorphic
representation of $\mathrm{GL}_{26}$ over $\mathbb{Q}$ with Sato-Tate group
$\mathbf{F}_{4}(\mathbb{R})$ of any given weight (assumed
"$\mathrm{F}_{4}$-regular"). The first example of such representations occurs
in motivic weight $36$.
