Isomorphisms of $5$-configurations obtained from $2$-digraphs

We consider $5$-configurations defined by their incident matrices over the field $\text{GF}(2)$, which must be nonsingular and contain exactly $5$ units in each row and each column, and the inverse matrix must also have this property. Automorphisms of $5$-configurations are studied. The relationship is shown between the group of automorphisms of an oriented graph without loops and parallel arcs with two input and two output arcs at each vertex and the group of automorphisms of the $5$-configuration obtained from this digraph.