Sylow normalizers and character tables. II. (Q1860711)

[For part I cf. \textit{I. M. Isaacs} and the author, Arch. Math. 78, No. 6, 430-434 (2002; Zbl 1036.20004).] If \(P\) is a Sylow \(p\)-subgroup of a \(p\)-solvable group \(G\), then the character table of \(G\) determines whether \(N_G(P)/P\) is Abelian. The arguments which lead to this result, heavily use a strong form of the Alperin-McKay conjecture for \(p\)-solvable groups which does not hold in general. Consequently, the present approach cannot decide whether the above result is true or false for every finite group. Some intermediate results are of independent interest. For example, every irreducible character of an Abelian normal subgroup \(N\) of \(G\), \(G'\leq N\), extends to \(G\) if and only if \(G\) is Abelian.