On the group of automorphisms of the Brandt λ⁰-extension of a monoid with zero

The group of automorphisms of the Brandt $\lambda^0$-extension $B^0_\lambda(S)$ of an arbitrary monoid $S$ with zero is described. In particular we show that the group of automorphisms $\mathbf{Aut}(B_{\lambda}^0(S))$ of $B_{\lambda}^0(S)$ is isomorphic to a homomorphic image of the group defines on the Cartesian product $\mathscr{S}_{\lambda}\times \mathbf{Aut}(S)\times H_1^{\lambda}$ with the following binary operation: \begin{equation*} [\varphi,h,u]\cdot[\varphi^{\prime},h^{\prime},u^{\prime}]= [\varphi\varphi^{\prime},hh^{\prime},\varphi u^{\prime}\cdot uh^{\prime}], \end{equation*} where $\mathscr{S}_{\lambda}$ is the group of all bijections of the cardinal $\lambda$, $\mathbf{Aut}(S)$ is the group of all automorphisms of the semigroup $S$ and $H_1^{\lambda}$ is the direct $\lambda$-power of the group of units $H_1$ of the monoid $S$.