Central translations in a linear semidirect product ($(e,v)\in Z(V⋊H)⟺\rho (h)v=v\text{for all }h$)

Let an abelian group $V$ carry an action $\rho$ of a group $H$ by automorphisms, and use multiplication $(h,v)(k,w)=(hk,v+\rho (h)w)$. Comparing $(e,v)(h,w)$ and $(h,w)(e,v)$ shows that a translation is central precisely when its vector is fixed by the entire action. If $H$ is itself abelian, the whole semidirect product is abelian exactly when the action is trivial. For $H=(\mathbb{R},+)$, $V={\mathbb{R}}^2$, and determinant-one linear actions fixing the first coordinate vector, $\rho (x)=\left(\begin{array}{cc}1& r(x)\\ 0& 1\end{array}\right)$, where $r$ is an additive function. Without a regularity assumption, it need not be $r(x)=cx$.