  We study $\kappa$-maximal cofinitary groups for $\kappa$ regular uncountable,
$\kappa = \kappa^{<\kappa}$. Revisiting earlier work of Kastermans and building
upon a recently obtained higher analogue of Bell's theorem, we show that:
  1. Any $\kappa$-maximal cofinitary group has ${<}\kappa$ many orbits under
the natural group action of $S(\kappa)$ on $\kappa$.
  2. If $\mathfrak{p}(\kappa) = 2^\kappa$ then any partition of $\kappa$ into
less than $\kappa$ many sets can be realized as the orbits of a
$\kappa$-maximal cofinitary group.
  3. For any regular $\lambda > \kappa$ it is consistent that there is a
$\kappa$-maximal cofinitary group which is universal for groups of size
${<}2^\kappa = \lambda$. If we only require the group to be universal for
groups of size $\kappa$ then this follows from $\mathfrak{p}(\kappa) =
2^\kappa$.
