  Let $G$ be a group endowed with a solution to the conjugacy problem and with
an algorithm which computes the centralizer in $G$ of any element of $G$. Let
$H$ be a subgroup of $G$. We give some conditions on $H$, under which we
provide a solution to the conjugacy problem in $H$. We apply our results to
some Artin groups and braid type groups. In particular, we give explicit
solutions to the conjugacy problem in the Artin groups of type $\tilde{A}_n$
and $\tilde{C}_n$.
