Decomposition of idempotent 2-cocycles

Let L be a finite Galois extension of K with Galois group G. We decompose any idempotent 2-cocycle f using finite sequences of descending two-sided ideals of the corresponding weak crossed product algebra A:= (L/k, G, f). We specialise the results in case f is the corresponding idempotent 2-cocycle for some semilinear map from G to a multiplicative monoid with minimum element.