Similarity and consimilarity automorphisms of the space of Toeplitz matrices

Let $T_n$ be the set of complex Toeplitz $n\times n$ matrices. We describe the matrices $U$ in the linear group $\mathrm{GL}_n(\mathbf{C})$ such that
$$ \forall A \in T_n \longrightarrow U^{-1}AU \in T_n $$
and the matrices $U \in \mathrm{GL}_n(\mathbf{C})$ such that
$$ \forall A \in T_n \longrightarrow U^{-1}A\bar U \in T_n. $$