Suppose we have a discrete group $G$, finite or infinite, on which we form the group algebra $\mathbb{F}_2[G]$. Suppose also that we have a map $S$:
$S : \mathbb{F}_2[G] \to \mathbb{F}_2[G]: \sum{a_g \cdot g} \to \sum{a_g \cdot g^{-1}}$
The problem I'm trying to solve is the following : given an element $x \in \mathbb{F}_2[G]$, find all $y \in \mathbb{F}_2[G]$ such that $y \cdot S(y) = x$.
For small groups $G$, we can write down a system of equations on the $a_g$ but I cannot find a general way... any help or lead appreciated !