  S. Gersten announced an algorithm that takes as input two finite sequences
$\vec K=(K_1,\dots, K_N)$ and $\vec K'=(K_1',\dots, K_N')$ of conjugacy classes
of finitely generated subgroups of $F_n$ and outputs:
  (1) $\mathsf{YES}$ or $\mathsf{NO}$ depending on whether or not there is an
element $\theta\in \mathsf{Out}(F_n)$ such that $\theta(\vec K)=\vec K'$
together with one such $\theta$ if it exists and
  (2) a finite presentation for the subgroup of $\mathsf{Out}(F_n)$ fixing
$\vec K$.
  S. Kalajd\v{z}ievski published a verification of this algorithm. We present a
different algorithm from the point of view of Culler-Vogtmann's Outer space.
  New results include that the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$
is of type $\mathsf{VF}$, an equivariant version of these results, an
application, and a unified approach to such questions.
