Decomposition of idempotent 2-cocycles

Let $L$ be a finite Galois field 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 $A_f$. We specialize the results in case $f$ is the corresponding idempotent 2-cocycle $f_r$ for some semilinear map $r:G\rightarrow \Omega $, where $\Omega $ is a multiplicative monoid with minimum element.