Surprising fun fact

I have just found a proof of the following. Usual caveat: nobody else has read the proof yet, and I have not carefully checked it.

Let G be a finite group. The finite group H will be called an inverse Frattini group of G if G is isomorphic to the Frattini subgroup of H. As the climax of a series of papers going back to Gaschütz, Bettina Eick proved a beautiful theorem: G has an inverse Frattini group if and only if the inner automorphism group of G is contained in the Frattini subgroup of the automorphism group of G.

Now here is my new fact.

It is certainly not true that a group has an inverse Frattini group if and only if it is abelian!

Part of the proof involves showing that, for every prime number p and positive integer m ≥ 3, there is a group of order p^m which has no inverse Frattini group. This is true, but computational evidence suggests that much more is true; I conjecture that the proportion of groups of order p^m with no inverse Frattini group tends to 1 as m tends to infinity, for fixed p. Here are the proportions for p = 2, m = 3,…,8: 0.4, 0.5, 0.62745…, 0.74906…, 0.87715…, 0.96213….