On the intersection of non-$\mathfrak{F}$-subgroups, distinguished by a subgroup function, in groups with operators

The structure of a subgroup associated with the intersection of kernels of maximal $A$-admissible subgroups that do not contain the $\mathfrak{F}$-residual and do not belong to the $\mathfrak{F}$ formation whose indices have restrictions are considered. The properties of the corresponding generalized Frattini subgroup are established.