On the embedding of a finite group as Frattini subgroup (Q1909615)

The authors investigate the titular groups of order 32, continuing the work of the first author on groups of order 16 [Indag. Math. 36, 382-386 (1974; Zbl 0288.20020)]. For all except groups \(\#16\) and \(\#40\) (in the Hall-Senior listing) it is determined exactly which ones occur as Frattini subgroups. The exceptional groups \(N\) are shown not to occur as Frattini subgroups of 2-groups but to have \(\text{Inn}(N)\leq\Phi(\text{Aut}(N))\). This means that one of the following ``folklore conjectures'' is false in general: 1. A \(p\)-group \(P\) is the Frattini subgroup of some group if and only if it is the Frattini subgroup of some \(p\)-group. 2. If \(\text{Inn}(N)\leq\Phi(\text{Aut}(N))\) for a group \(N\) then \(N\) is the Frattini subgroup of some group \(G\).