NSCFN-realizable group

Definition
A group $$G$$ is termed NSCFN-realizable if it satisfies the following equivalent conditions:


 * $$G$$ can be embedded as a NSCFN-subgroup of some group.
 * Given any homomorphism $$\alpha: L \to \operatorname{Out}(G)$$, there exists a group $$K$$ containing $$G$$ as a normal subgroup with $$K/G$$ isomorphic to $$L$$, and the induced outer action of $$G$$ on $$L$$ is $$\alpha$$.
 * The outer action cohomology class in $$H^3(\operatorname{Out}(G),Z(G))$$ is trivial.

Stronger properties

 * Weaker than::Group whose center is a direct factor
 * Weaker than::Abelian group
 * Weaker than::Centerless group
 * Weaker than::Group in which every automorphism is inner

Facts

 * Special linear group:SL(2,9) is not NSCFN-realizable