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, Theory Appl. Categ. 12, 423-491 (2004). ZBL1056.18002.
It is also known 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.
