TITLE: Groups all of whose extensions are split
QUESTION [8 upvotes]: Is there a sensible characterization of groups $G$ with the following property?

Every extension of groups $1\to G\to H\to K\to 1$ is split.

A complete group $G$ has that property and in fact such a group has a normal complement in every group that contains it as a normal subgroup (moreover, completeness is characterized by this)
For comparison, a group $K$ has the property that every extension $1\to G\to H\to K\to 1$ is split iff it is free (there is such an extension with $H$ free,  and the splitting map $K\to H$ is injective, so gives an isomorphism of $K$ with a subgroup of $H$, which is free by the Nielsen–Schreier theorem)
NB: the title does use the old meaning of «extension of a group»...

REPLY [5 votes]: Since my late comment to YCor's answer is easily overlooked, I allow myself to repeat it here: The question was answered in [J. S. Rose, Splitting properties of group extensions, Proc. London Math. Soc. (3) 22 (1971), 1–23] with exactly the same outcome as in YCor's answer.