  The study of automorphisms of computable and other structures connects
computability theory with classical group theory. Among the noncomputable
countable structures, computably enumerable structures are one of the most
important objects of investigation in computable model theory. In this paper,
we focus on the lattice structure of computably enumerable substructures of a
given canonical computable structure. In particular, for a Turing degree
$\mathbf{d}$, we investigate the groups of $\mathbf{d}$ -computable
automorphisms of the lattice of $\mathbf{d}$-computably enumerable vector
spaces, of the interval Boolean algebra $\mathcal{B}_{\eta }$ of the ordered
set of rationals, and of the lattice of $\mathbf{d}$ -computably enumerable
subalgebras of $\mathcal{B}_{\eta }$. For these groups we show that Turing
reducibility can be used to substitute the group-theoretic embedding. We also
prove that the Turing degree of the isomorphism types for these groups is the
second Turing jump of $\mathbf{d}$ , $\mathbf{d^{\prime \prime }}$.
