  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 $\Omega$.
