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We define the Frobenius limit of a module over a ring of prime characteristicto be the limit of the normalized Frobenius direct images in a certainGrothendieck group. When a finite group acts on a polynomial ring, we calculatethis limit for all the modules over the twisted group algebra that are freeover the polynomial ring; we also calculate the Frobenius limit for therestriction of these to the ring of invariants. As an application, wegeneralize the description of the generalized $F$-signature of a ring ofinvariants by the second author and Nakajima to the modular case.

The asymptotic behavior of Frobenius direct images of rings of invariants