  Let $G$ be a residually finite, good group of finite virtual cohomological
dimension. We prove that the natural monomorphism $G\hookrightarrow\hat{G}$
induces a bijective correspondence between conjugacy classes of finite
$p$-subgroups of $G$ and those of its profinite completion $\hat{G}$. Moreover,
we prove that the centralizers and normalizers in $\hat{G}$ of finite
$p$-subgroups of $G$ are the closures of the respective centralizers and
normalizers in $G$. With somewhat more restrictive hypotheses, we prove the
same results for finite solvable subgroups of $G$. In the last section, we give
a few applications of this theorem to hyperelliptic mapping class groups and
virtually compact special toral relatively hyperbolic groups (these include
fundamental groups of $3$-orbifolds and of uniform standard arithmetic
hyperbolic orbifolds).
