  J. Pevtsova and the author constructed a ``universal $p$-nilpotent operator"
for an infinitesimal group scheme $G$ over a field $k$ of characteristic $p >
0$ which led to coherent sheaves on the scheme of 1-parameter subgroups of $G$
associated to a $G$-module $M$. Of special interest is the fact that these
coherent sheaves are vector bundles if $M$ is of constant Jordan type. In this
paper, we provide similar invariants for a finite group $\tau$ which recover
the invariants earlier obtained for elementary abelian $p$-groups. To do this,
we replace the analogue of 1-parameter subgroups by a refined version of
equivalence classes of $\pi$-points for $k\tau$. More generally, we provide a
construction of vector bundles for the semi-direct product $G\rtimes \tau$ of
an infinitesimal group scheme $G$ and a finite group $\tau$.
  A major motivation for this study is to further our understanding of the
relationship between representations of $\mathbb G(\mathbb F_p)$ and $\mathbb
G_{(r)}$ associated to a finite dimensional rational $\mathbb G$-module $M$,
where $\mathbb G$ is a reductive group with $r$-th Fobenius kernel $\mathbb
G_{(r)}$. Using vector bundles, we extend and sharpen earlier results comparing
support varieties.
