Fact 1 (see this MSE post): The semidirect product $G \rtimes H$ is abelian iff $G$ is abelian, $H$ is abelian, and the semidirect product is trivial (and thus is just a product.)
So we may ask, which nonabelian groups are nontrivial semidirect products of two abelian groups?
The general problem of understanding how groups decompose under the semidirect product is hard. The only related theorem I know of is the Schur-Zassenhaus theorem. I do not think it directly gives any answers to my question.
I am interested in and am here to ask for partial results or observations related to my question. E.g., can any reasonable restrictions on our non-abelian group rule it out as a SDP of abelian groups?
I know of no such classification results and am unable to find any online. Keith Conrad has some notes discussing when a group decomposes into a SDP of subgroups, but this is not the same.