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11.26 (1990)

Solved

Does there exist a group which is not isomorphic to outer automorphism group of a metabelian group with trivial center?

Contributor: R. Göbel

Progress

No, given any group $G$ there is a metabelian group $M$ with trivial center such that $\text{Out}\,M \cong G$ (R. Göbel, A. Paras, J. Pure Appl. Algebra, 149, no. 3 (2000) 251–266), and if $G$ is finite or countable then $M$ above can be chosen countable (R. Göbel, A. Paras, in: Abelian Groups and Modules, Proc. Int. Conf. Dublin, 1998, Birkhäuser, Basel, 1999, 309–317).

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