  A $nil$-closed, noetherian, unstable algebra $K$ over the Steenrod Algebra is
determined, up to isomorphism, by the functor
$\text{Hom}_{\mathcal{K}\text{f.g.}}(K,H^*(\_))$, which is a presheaf on the
category $\mathcal{VI}$ of finite dimensional vector spaces and injections, by
the theory of Henn-Lannes-Schwartz. In this article, we use this theory to
study the centre, in the sense of Heard, of a $nil$-closed noetherian unstable
algebra. For $F$ a presheaf on $\mathcal{VI}$, we construct a groupoid
$\mathcal{G}_F$ which encodes $F$. Then, taking
$F:=\text{Hom}_{\mathcal{K}\text{f.g.}}(K,H^*(\_))$, we show how the centre of
$K$ is determined by the associated groupoid. We also give a generalisation of
the second theorem of Adams-Wilkerson, defining sub-algebras
$H^*(W)^\mathcal{G}$ of $H^*(W)$ for appropriate groupoids $\mathcal{G}$. There
is a $H^*(C)$-comodule structure on $K$ that is associated with the centre. For
$K$ integral, we explain how the algebra of primitive elements of this
$H^*(C)$-comodule structure is also determined by the groupoid associated with
$\text{Hom}_{\mathcal{K}\text{f.g.}}(K,H^*(\_))$. Along the way, we prove that
this algebra of primitive elements is also noetherian.
