  We describe a strategy for the construction of finitely generated
$G$-equivariant $\mathbb{Z}$-graded modules $M$ over the exterior algebra for a
finite group $G$. By an equivariant version of the BGG correspondence, $M$
defines an object $\mathcal{F}$ in the bounded derived category of
$G$-equivariant coherent sheaves on projective space. We develop a necessary
condition for $\mathcal{F}$ being isomorphic to a vector bundle that can be
simply read off from the Hilbert series of $M$. Combining this necessary
condition with the computation of finite excerpts of the cohomology table of
$\mathcal{F}$ makes it possible to enlist a class of equivariant vector bundles
on $\mathbb{P}^4$ that we call strongly determined in the case where $G$ is the
alternating group on $5$ points.
