Suppose $G$ is a finite centerless metabelian group. Is it true that it is a semidirect product of two abelian groups?
It does not seem true to me, but I failed to find any counterexamples. Actually, I know quite a few examples of finite metabelian groups that do not split into a semidirect product of two abelian groups, but all of them have a nontrivial centre.
What I have tried:
The only thing I managed to see here is that a group having an abelian normal Hall subgroup with an abelian quotient by it is a semidirect product of two abelian groups (as any finite group splits over its normal Hall subgroup). On the other hand, all Hall subgroups of a semidirect product of two finite abelian groups are normal (as Hall subgroups of finite abelian groups are always characteristic).
So that question can be reduced to either finding a finite centerless metabelian group with a non-normal Hall subgroup or proving that any finite centerless metabelian group has an abelian normal Hall subgroup with an abelian quotient by it. Note that aforementioned implications are one-sided, so proving that all Hall subgroups of finite centerless metabelian groups are normal or finding a finite centerless metabelian group without a normal abelian Hall subgroup with an abelian quotient by it will not give us anything regarding this question.
However, those two new "questions" do not seem any easier than the initial one...