Controlling LEF growth in some group extensions

For a finitely generated LEF group $\Gamma$ , we study the orders of finite groups admitting local embeddings of balls in a word metric on $\Gamma$ , as measured by the LEF growth function. We prove that any sufficiently smooth increasing function between n! and $exp(exp(n\left)\right)$ 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 $\text{FSym}\left(\Omega \right)⋊\Gamma$ , where $\Omega \curvearrowleft \Gamma$ is an appropriate transitive action and $\text{FSym}\left(\Omega \right)$ 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 $E_{\Omega }\left(R\right)⋊\Gamma$ , for R an appropriate unital ring and $E_{\Omega }\left(R\right)$ the subgroup of ${\text{Aut}}_R\left(R\left[\Omega \right]\right)$ generated by all transvections with respect to basis $\Omega$ .