A McCool Whitehead type theorem for finitely generated subgroups of $mathsf{Out}(F_n)$

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 $thetain mathsf{Out}(F_n)$ 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. Kalajdv{z}ievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler-Vogtmanns 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.