  Let $E$ be a $W^{*}$-correspondence and let $H^{\infty}(E)$ be the associated
Hardy algebra. The unit disc of intertwiners $\mathbb{D}((E^{\sigma})^{*})$
plays a central role in the study of $H^{\infty}(E)$. We show a number of
results related to the automorphism groups of both $H^{\infty}(E)$ and
$\mathbb{D}((E^{\sigma})^{*})$. We find a matrix representation for these
groups and describe several features of their algebraic structure. Furthermore,
we show an application of $Aut(\mathbb{D}({(E^{\sigma}})^*))$ to the study of
Morita equivalence of $W^{*}$-correspondences.
