A group $G$ has the structure of an inner semidirect product when it can be reconstructed from two of its subgroups: one, often written $N \subset G$, is a normal subgroup, and the other one, $H \subset G$ is a complement subgroup such that $G = \{ n h : n \in N, h \in H \}$.
In all the materials I've read, those subgroups do not have a particular name. However, I want to give them explicit names in a MATLAB library I'm writing. Are you aware of a particular terminology here? For the inner semidirect product, I could use "normal subgroup" and "complement subgroup".
However, I'm dealing with outer semidirect products, where a group $G$ is composed of pairs $(n,h) \in N \times H$ with a particular group operation. How could I call the groups $N$ and $H$ involved there?
Some standard terminology is:
If $G=N\rtimes H$ then $H$ is a retract of $G$ with normal complement $N$.