  We define the Frobenius limit of a module over a ring of prime characteristic
to be the limit of the normalized Frobenius direct images in a certain
Grothendieck group. When a finite group acts on a polynomial ring, we calculate
this limit for all the modules over the twisted group algebra that are free
over the polynomial ring; we also calculate the Frobenius limit for the
restriction of these to the ring of invariants. As an application, we
generalize the description of the generalized $F$-signature of a ring of
invariants by the second author and Nakajima to the modular case.
