Supergroups of alternating group:A6

This article discusses some of the groups that admit the alternating group of degree six as a subgroup, quotient group, or subquotient.

Note that unlike the discussion of the subgroup structure of alternating group:A6, this discussion is necessarily not comprehensive, because there are infinitely many groups containing the alternating group of degree four.

Subgroups: making some or all the outer automorphisms inner
The outer automorphism group of alternating group:A6 is a Klein four-group. In particular, it has order 4. By the fourth isomorphism theorem, subgroups of the automorphism group containing the inner automorphism group correspond to subgroups of the outer automorphism group, which is the quotient group of the automorphism group by the inner automorphism group.

Since alternating group:A6 is a centerless group, it is identified naturally with its inner automorphism group, so each of the subgroups of the automorphism group containing the inner automorphism group is also a group containing $$A_6$$ as a self-centralizing normal subgroup. The whole automorphism group contains $$A_6$$ as a NSCFN-subgroup.

Below is the complete list of these groups.:

Quotients: Stem extensions and Schur covering groups
The Schur multiplier of alternating group:A6 is cyclic group:Z6, i.e., the group $$\mathbb{Z}/6\mathbb{Z}$$. For each of the possible quotient groups of $$\mathbb{Z}/6\mathbb{Z}$$, there is a unique stem extension with that as base normal subgroup and alternating group:A6 as quotient. The stem extension for the whole Schur multiplier is the unique Schur covering group, also called the universal central extension.

Note that uniqueness follows from $$A_6$$ being a perfect group.

The list is below: