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 \(\exp (\exp (n))\) 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 \({{\,\textrm{FSym}\,}}(\Omega ) \rtimes \Gamma \) , where \(\Omega \curvearrowleft \Gamma \) is an appropriate transitive action and \({{\,\textrm{FSym}\,}}(\Omega )\) 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 \({{\,\textrm{Aut}\,}}_R (R[\Omega ])\) generated by all transvections with respect to basis \(\Omega \) .
