On the problem of Doerk and Hawkes for locally normal Fitting classes

Let $\mathfrak{F}$ be a non-empty class of finite groups. A Fitting class $\mathfrak{F}$ is said to be $\mathfrak{X}$-normal or normal in a class of finite groups $\mathfrak{X}$ if $\mathfrak{F}\subseteq\mathfrak{X}$ and for all $G\in\mathfrak{X}$ an $\mathfrak{F}$-radical of $G$ is $\mathfrak{F}$-maximal in $G$. If $\mathfrak{X}$ is a class of all soluble finite groups, then $\mathfrak{X}$-normal Fitting class is called normal. In the theory of normal Fitting classes the problem of Doerk and Hawkes is well known. Let $\mathfrak{X}$ be a Fitting class and $\mathfrak{X}=\mathfrak{X}^2$. Is the intersection of two non-trivial $\mathfrak{X}$-normal Fitting classes always non-trivial $\mathfrak{X}$-normal Fitting class? In this paper a positive answer to this question without the requirement that $\mathfrak{X}=\mathfrak{X}^2$ for the case of arbitrary family of non-trivial $\mathfrak{X}$-normal Fischer classes partially soluble groups, where $\mathfrak{X}$ is a Fischer class such, that $\mathfrak{N}_p\mathfrak{X}=\mathfrak{X}$ for some prime $p$ is given.