Extendable Outer Automorphisms of Subgroup Inclusions

Definitions and mathematical statement

Let $G$ be a group and $H\leq G$ a subgroup. Write $\operatorname{Aut}(H)$ for the automorphism group of $H$, $\operatorname{Inn}(H)$ for its subgroup of inner automorphisms, and \[ \operatorname{Out}(H)=\operatorname{Aut}(H)/\operatorname{Inn}(H). \] Define \[ \operatorname{Aut}(G;H)=\{\Phi\in\operatorname{Aut}(G):\Phi(H)=H\}. \] Restriction to $H$, followed by passage to outer automorphism classes, gives a homomorphism \[ \operatorname{Aut}(G;H)\longrightarrow\operatorname{Out}(H), \qquad \Phi\longmapsto[\Phi|_H]. \] The group of extendable outer automorphisms is its image: \[ \operatorname{Out}(H\hookrightarrow G) =\left\{[\alpha]\in\operatorname{Out}(H): \exists\Phi\in\operatorname{Aut}(G),\ \Phi(H)=H,\ \Phi|_H=\alpha\right\}. \] Here extendability means that an automorphism representative extends to an automorphism of $G$. This condition is independent of the representative: conjugation by an element of $H$ extends to conjugation by the same element of $G$.

Problem. For a subgroup inclusion $H\leq G$, describe the elements of $\operatorname{Out}(H\hookrightarrow G)$, and determine whether this group is finitely generated or finitely presented. Finite generation means that some finite subset generates the group; finite presentability means that the group admits a presentation with finitely many generators and finitely many defining relations.

Unresolved scope. The supplied statement imposes no finiteness, geometric, or other hypotheses on $G$ or $H$. It also does not specify what kind of description is intended by "what does this group contain." Thus the statement records a general characterization and finiteness question, without asserting a universal affirmative answer or specifying a narrower class of inclusions.

Short English statement

For a subgroup $H\leq G$, consider the outer automorphism classes of $H$ represented by restrictions of automorphisms of $G$ preserving $H$. Describe this subgroup of $\operatorname{Out}(H)$ and investigate its finite generation and finite presentability. The supplied question leaves the classes of groups and the intended form of the description unspecified.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30002256$, source problem code catalogue entry, title Extendable Outer Automorphisms of Subgroup Inclusions; registry locations data/unsolvedmath/problems.json and catalog/problems.json. The record assigns the year 2013 and supplies no individual proposer.
• Model Theory: Groups, Geometry, and Combinatorics (2013), Oberwolfach Reports, pp. 5--66, DOI 10.4171/owr/2013/01: https://doi.org/10.4171/owr/2013/01. Also listed in the supplied literature sources as Model Theory, Oberwolfach Report 01/2013. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Splittings and Automorphisms of Relatively Hyperbolic Groups: https://arxiv.org/abs/1212.1434. The supplied assessment attributes to Guirardel and Levitt a description of the extendable subgroup and finite-index McCool and finite-presentation results for malnormal quasiconvex and relatively hyperbolic cases, but not for arbitrary subgroup inclusions. The precise hypotheses and conclusions of these results are not included in the record.
• Literature assessment date: 2026-08-21. The supplied dated curation labels the problem partially_solved. This assessment, including the attributed literature claims, is not independently verified here; the source itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with a supplied verification note saying that the official report states the general question with the same extendability definition and finiteness alternatives. This is registry-provided source verification, not an independent verification performed for this draft.