On pyramidal groups of prime power degree

A Kirkman Triple System Γ is called m-pyramidal if there exists a subgroup G of the automorphism group of Γ that fixes m points and acts regularly on the other points. Such group G admits a unique conjugacy class C of involutions (elements of order 2) and |C|=m. We call groups with this property m-pyramidal. We prove that, if m is an odd prime power pk, with p≠7, then every m-pyramidal group is solvable if and only if either m=9 or k is odd. The primitive permutation groups play an important role in the proof. We also determine the orders of the m-pyramidal groups when m is a prime number.