  In 1971 C.F.\ Miller associated to every finitely presented group $G$ a
free-by-free group $M(G)$ known as the Miller Machine, whose conjugacy problem
is closely related to the conjugacy and word problems of $G$. We quantify this
relationship, and look to fully understand the conjugacy problem of $M(G)$;
namely, we reduce the conjugacy problem in $M(G)$ to a strong form of list
conjugacy in $G$, which we term iso-computational list conjugacy.
