  In this paper, we characterize the extremal digraphs with the maximal or
minimal $\alpha$-spectral radius among some digraph classes such as rose
digraphs, generalized theta digraphs and tri-ring digraphs with given size $m$.
These digraph classes are denoted by $\mathcal{R}_{m}^k$,
$\widetilde{\boldsymbol{\Theta}}_k(m)$ and $\INF(m)$ respectively. The main
results about spectral extremal digraph by Guo and Liu in \cite{MR2954483} and
Li and Wang in \cite{MR3777498} are generalized to $\alpha$-spectral graph
theory. As a by-product of our main results, an open problem in
\cite{MR3777498} is answered.
  Furthermore, we determine the digraphs with the first three minimal
$\alpha$-spectral radius among all strongly connected digraphs. Meanwhile, we
determine the unique digraph with the fourth minimal $\alpha$-spectral radius
among all strongly connected digraphs for $0\le \alpha \le \frac{1}{2}$.
