20.122 (2022)

For nilpotent subgroups $A, B, C$ of a finite group $G$, let $\text{Min}_G(A, B, C)$ be the subgroup of $A$ generated by all minimal by inclusion intersections of the form $A \cap B^x \cap C^y$, where $x, y \in G$, and let $\text{min}_G(A, B, C)$ be the subgroup of $\text{Min}_G(A, B, C)$ generated by all intersections of this kind of minimal order.
$\qquad$ a) Is it true that $\text{min}_G(A, B, C) \leqslant F(G)$?
$\qquad$ b) Is it true that $\text{Min}_G(A, B, C) \leqslant F(G)$?
$\qquad$ c) The same questions for soluble groups.