Controlling LEF growth in some group extensions

We study the LEF growth function of a finitely generated LEF group $\Gamma$, which measures the orders of finite groups admitting local embeddings of balls in a word metric on $\Gamma$. We prove that any sufficiently smooth increasing function between $n!$ and $\exp(\exp(n))$ is close to the LEF growth function of some finitely generated group. This is achieved by estimating the LEF growth of some semidirect products of the form $FSym (\Omega) \rtimes \Gamma$, where $\Gamma \curvearrowright \Omega$ is an appropriate transitive action, and $FSym (\Omega)$ is the group of finitely supported permutations of $\Omega$. A key tool in the proof is to identify sequences of finitely presented subgroups with short "relative" presentations. In a similar vein we also obtain estimates on the LEF growth of some groups of the form $E_{\Omega} (R) \rtimes \Gamma$, for $R$ an appropriate unital ring and $E_{\Omega} (R)$ the subgroup of $Aut_R (R[\Omega])$ generated by all transvections with respect to basis $\Omega$.