On inner automorphisms and certain central automorphisms of groups. (Q2018721)

Let \(G\) be a group and \(M,N\trianglelefteq G\). By definition an automorphism \(\alpha\) of \(G\) belongs to \(\Aut^M_N(G)\) if and only if \(g^{-1}g^\alpha\in M\) for all \(g\in G\) and \(\alpha\) fixes \(N\) elementwise. The paper under review is devoted to the study of groups \(G\) in which one of the following holds: \(\mathrm{Inn}(G)=\Aut^M_N(G)\), \(\mathrm{Inn}(G)\leq\Aut^M_N(G)\) or \(\mathrm{Inn}(G)\geq\Aut^M_N(G)\).