Given a group $G$, the outer automorphism group $Out(G)$ acts on the center by $Z(G)$ by lifting an outer automorphism to an actual automorphism and evaluating this on elements of $Z(G)$. What is classified by the degree three group cohomology $H^3(Out(G),Z(G))$ ?


Given an algebra $A$, say finite-dimensional over a field, the outer automorphism group $Out(A)$ acts on the group of central units $Z(A)^\times$ by lifting an outer automorphism to an actual automorphism and then evaluating. The action is trivial if $A$ is a central algebra. What is classified by the degree three group cohomology $H^3(Out(A),Z(A)^\times)$ ?


My motivation for this question is the following. If $A$ is my algebra, then we have a *crossed module*
$$
A^\times \to Aut(A)
$$
given by inner automorphisms and the evaluation action of $Aut(A)$ on $A^\times$. The homotopy groups of this crossed module are $\pi\_0=Out(A)$ and $\pi\_1=Z(A)^\times$, and the usual action of $\pi\_0$ on $\pi\_1$ is the one described above. Crossed modules have a so-called *k-invariant*, which is precisely a class $\xi\in H^3(\pi\_0,\pi\_1)$.


Baez and Lauda have shown that crossed modules are classified in a sense by triples $(\pi\_0,\pi\_1,\xi)$. The classification is saying that the crossed module - viewed as a monoidal category - is equivalent to the usual monoidal category constructed from the 3-cocycle $\xi$.


*Baez, John C.; Lauda, Aaron D.*, [**Higher-dimensional algebra. V: 2-Groups**](http://www.emis.de/journals/TAC/volumes/12/14/12-14abs.html), Theory Appl. Categ. 12, 423-491 (2004). [ZBL1056.18002](https://zbmath.org/?q=an:1056.18002).


It is also [known](https://arxiv.org/abs/2204.03900) that if $A$ and $B$ are Picard-surjective and Morita equivalent, then their crossed modules are equivalent and so their classes coincide. Moreover, if $A$ is Picard-surjective and central-simple, then its class vanishes.


Summarizing, associated to every algebra is a class in $H^3(Out(A),Z(A)^\times)$. What is the intrinsic meaning of this class, apart from classifying some crossed module?


For groups instead of algebras it is basically the same story, and the question is analogous.



3-cocycles on outer automorphism groups
Among other things, the third cohomology contains an invariant for the existence of group-graded algebras whose degree-1-piece is $A$ / group extension with $G$ as normal subgroup. This is a theorem of Schreier. If I'm not mistaken, the cohomology class $\xi$ that comes from crossed modules is precisely Schreier's invariant.


There is a generalisation of both of these with more abstract nonsense sprinkled in (2-groups among them) see [this earlier question of mine](https://mathoverflow.net/questions/401964/connection-between-the-classifications-of-group-extensions-and-group-graded-alge)


