On the $\mathfrak{F}$-hypercentral subgroups with the sylow tower property of finite groups

Throughout this paper all groups are finite. Let $A$ be a group of automorphisms of a group $G$ that contains all inner automorphisms of $G$ and $F$ be the canonical local definition of a saturated formation $\mathfrak{F}$. An $A$-composition factor $H/K$ of $G$ is called $A$-$\mathfrak{F}$-central if $A/C_A(H/K)\in F(p)$ for all $p\in\pi(H/K)$. The $A$-$\mathfrak{F}$-hypercenter of $G$ is the largest $A$-admissible subgroup of $G$ such that all its $A$-composition factors are $A$-$\mathfrak{F}$-central. Denoted by $\mathrm{Z}_\mathfrak{F}(G, A)$.
Recall that a group $G$ satisfies the Sylow tower property if $G$ has a normal Hall $\{p_1,\dots, p_i\}$-subgroup for all $1\leq i\leq n$ where $p_1>\dots>p_n$ are all prime divisors of $|G|$. The main result of this paper is: Let $\mathfrak{F}$ be a hereditary saturated formation, $F$ be its canonical local definition and $N$ be an $A$-admissible subgroup of a group $G$ where $\mathrm{Inn}\,G\leq A\leq \mathrm{Aut}\,G$ that satisfies the Sylow tower property. Then $N\leq\mathrm{Z}_\mathfrak{F}(G, A)$ if and only if $N_A(P)/C_A(P)\in F(p)$ for all Sylow $p$-subgroups $P$ of $N$ and every prime divisor $p$ of $|N|$.
As corollaries we obtained well known results of R. Baer about normal subgroups in the supersoluble hypercenter and elements in the hypercenter.
Let $G$ be a group. Recall that $L_n(G)=\{ x\in G\,\,| \,\,[x, \alpha_1,\dots, \alpha_n]=1 \,\,\forall \alpha_1,\dots, \alpha_n\in\mathrm{Aut}\,G\}$ and $G$ is called autonilpotent if $G=L_n(G)$ for some natural $n$. The criteria of autonilpotency of a group also follow from the main result. In particular, a group $G$ is autonilpotent if and only if it is the direct product of its Sylow subgroups and the automorphism group of a Sylow $p$-subgroup of $G$ is a $p$-group for all prime divisors $p$ of $|G|$. Examples of odd order autonilpotent groups were given.