Let $K=F_2$ and $A \in F_2^{n \times n}\quad$ which is the adjacency matrix of an arbitrary graph. We want to determine what permutation cycles are embedded in $A$, for that select $F_2G$ where $\vert G \vert=n$.
A natural way is to think about the adjacency matrix decomposition in $RG$ as it could be formed by linear combinations with coefficients on $F_2$ and basis elements are the permutation matrices of $G$. Let $\phi : RG \mapsto GL(n,F_2)$ be the morphism that sends an element $g_i$ in $RG$ to its matrix representation in $GL$.
$$A = \sum_{i=1}^{n}\alpha_i \phi(g_i)$$
It results that the equations can be reordered to form a linear equation system. $$B \in F_2^{n^2 \times n}, \quad x=(\alpha_1, \cdots, \alpha_n) \in F_2^{n \times 1}, \quad b \in F_2^{n^2 \times 1}$$
$$B = \begin{pmatrix} \phi(g_1) _{1,1}& \cdots & \phi(g_n)_{1,1}\\ \phi(g_1)_{1,2} & \cdots & \phi(g_n) _{1,2}\\ \cdots & \cdots & \cdots\\ \phi(g_1) _{n,n}& \cdots & \phi(g_n)_{n,n}\\ \end{pmatrix}, \quad b = \begin{pmatrix} A_{1,1} \\ \cdots \\ \cdots\\\cdots\\ A_{n,n} \end{pmatrix}$$
Think of $B$ as the matrix that is formed by stacking matrix representations of basis elements of $RG$ from top to bottom.
It's obvious that this system is overdetermined, furthermore, it only has a solution iff $A$ has a concrete structure, this is, being in $RG$.
One would think of how many matrices does $\phi$ form and how many matrices $A$ are out there.
$$\vert F_2G \vert = 2^n$$ $$\vert M_n(Adj) \vert = 2^{n^2}$$
Thus the proportion of having an adjacency matrix of binary values in $RG$ is small. Considering that I haven't misstaken in any part, which I hope is the case, I would want to ask the following questions:
Questions
Can this method be optimized or expanded to increase the quantity of arbitrary adj matrix that can be targeted?
What would be the best scenario to apply this method?