  A Kirkman triple system of order $v$, KTS$(v)$, is a resolvable Steiner
triple system on $v$ elements. In this paper, we investigate an open problem
posed by Doug Stinson, namely the existence of KTS$(v)$ which contain as a
subdesign a Steiner triple system of order $u$, an STS$(u)$. We present several
different constructions for designs of this form. As a consequence, we
completely settle the extremal case $v=2u+1$, for which a list of possible
exceptions had remained for close to 30 years. Our new constructions also
provide the first infinite classes for the more general problem. We reduce the
other maximal case $v=2u+3$ to now three possible exceptions. In addition, we
obtain results for other cases of the form $v=2u+w$ and also near $v=3u$. Our
primary method introduces a new type of Kirkman frame which contains group
divisible design subsystems. These subsystems can occur with different
configurations, and we use two different varieties in our constructions.
