On pyramidal groups whose number of involutions is a prime power

A Kirkman Triple System $\Gamma$ is called $m$-pyramidal if there exists a subgroup $G$ of the automorphism group of $\Gamma$ 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 $p^k$, with $p \neq 7$, 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.