  Let $X_{t}$ and $Y_{t}$ be two Markov chains, on state spaces $\Omega \subset
\hat{\Omega}$. In this paper, we discuss how to prove bounds on the spectrum of
$X_{t}$ based on bounds on the spectrum of $Y_{t}$. This generalizes work of
Diaconis, Saloff-Coste, Yuen and others on comparison of chains in the case
$\Omega = \hat{\Omega}$. The main tool is the extension of functions from the
smaller space to the larger, which allows comparison of the entire spectrum of
the two chains. The theory is used to give quick analyses of several chains
without symmetry. The main application is to a `random transposition' walk on
derangements.
