  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
$p^k$, 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.
