Find the Automorphism group of a Brandt semigroup $B(G,2)$ ,where $G $ is a cyclic group of order 4.
Take $G=\{ e, a ,a^2, a^3\}$
$B(G,2) = \{ (i,a^s , j) : 1 \leq i,j \leq 2 \ \ , 0\leq s \leq 3 \} \cup \{0\}$ and the binary operation is defind by $$(i, a^r , j) (k,a^s , l) = \begin{cases} (i,a^ra^s,l) & \text{if } \ \ j=k \\ 0 &\text{if } otherwise \end{cases}$$
As Benjamin Steinberg remarked, this special case is not very complicated and does not need my answer. However the automorphism group of $B(S,n)$, where $S$ is a monoid and $n\in\mathbb{N}$, was completely determined in Gutik (see Theorem 2).