Suppose $G_1=N_1\rtimes H_1$, $G_2=N_2\rtimes H_2$ are semidirect products of (not necessarily finite or abelian) groups. Suppose $f:N_1\to N_2$ and $g: H_1\to H_2$ are group homomorphisms.
How can we use this data to construct a morphism $h:G_1\to G_2$? The obvious map $(f,g)$ doesn't work since it does not preserve multiplication. It seems one needs to cleverly choose how to glue these such that in the end the necessary diagram commutes, but it is continuing to evade me.
Context is the following: suppose $N$ and $H$ are affine group schemes where $H$ acts on $N$ via group automorphisms. I wish to show that $G(R):=N(R)\rtimes H(R)$ yields an affine group scheme. In particular, I am struggling to see where $G$ takes morphisms of $k$-algebras $R\to S$.
Converting my comments into an answer: in the setup you describe there's an extra compatibility coming from the naturality of the action map $H(-) \times N(-) \to N(-)$, and this extra compatibility lets you write down $h$ componentwise in the obvious way.