  A combinatorial block design $D$ is called $3$-pyramidal if there exists a
subgroup $G$ of $\mbox{Aut}(D)$ fixing $3$ points and acting regularly on the
other points. If this happens, we say that the design is $3$-pyramidal under
$G$. In case $D$ is a Kirkman triple system, it is known that such a group $G$
has precisely $3$ involutions, all conjugate to each other. In this paper, we
obtain a classification of the groups with this property.
