We study the automorphism groups of the lattice of $\mathbf{d}$-computably
enumerable substructures of effective vector spaces and certain effective
Boolean algebras. Here, $\mathbf{d}$ is a Turing degree. We establish the
equivalence of the embedding relation for these automorphism groups with the
order relation of the corresponding Turing degrees. By a result of Guichard,
the automorphisms of the lattice of $\mathbf{d}$-computably enumerable
vector spaces are induced by $\mathbf{d}$-computable invertible semilinear
transformations, GSL$_{\mathbf{d}}$. We prove that the Turing degree
spectrum of the group GSL$_{\mathbf{d}}$ is greater or equal than the second jump of $\mathbf{d}$. We obtain a similar result in the context of effective Boolean algebras.
This is joint work with Rumen Dimitrov and Andrei Morozov.