Summary: 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 \(\text{Z}_\mathfrak{F}(G, A)\).
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 \(\operatorname{Inn} G\leq A\leq \operatorname{Aut}G\) that satisfies the Sylow tower property. Then \(N\leq\text{Z}_\mathfrak{F}(G, A)\) 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 \(L_n(G)=\{ x\in G\mid [x, \alpha_1,\dots, \alpha_n]=1 \forall \alpha_1,\dots, \alpha_n\in\operatorname{Aut}G\}\) 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.