  Let $q\geq 5$ be a prime power. In this note, we prove that if a plane curve
$\mathcal{X}$ of degree $q - 1$ defined over $\mathbb{F}_q$ without
$\mathbb{F}_q$-linear components attains the Sziklai upper bound $(d-1)q+1 = (q
- 1)^2$ for the number of its $\mathbb{F}_q$-rational points, then
$\mathcal{X}$ is projectively equivalent over $\mathbb{F}_q$ to the curve $
\mathcal{C}_{(\alpha,\beta,\gamma)} : \alpha X^{q - 1} + \beta Y^{q - 1} +
\gamma Z^{q - 1} = 0$ for some $\alpha, \beta, \gamma \in \mathbb{F}_q^{*}$
such that $\alpha + \beta + \gamma = 0$. This completes the classification of
curves that are extremal with respect to the Sziklai bound.
  Also, since the Sziklai bound is equal to the St\"ohr-Voloch's bound for
plane curves of degree $q - 1$, our main result classifies the
$\mathbb{F}_q$-Frobenius classical extremal plane curves of degree $q - 1$.
