  This paper explores a previously uncharted automaton group generated by a
5-state automaton $(\Pi, A)$ acts by self-similarity on the regular rooted tree
$A^{*}$ over a 2-letter alphabet set $A$. Group $G$ has been subjected to
several observations that reveal the self-similar $G$-action possesses several
notable characteristics: it is a weak branch, contracting, fractal group, and
satisfies the open set condition with an exponential growth rate in activity.
The corresponding Schreier graphs $\Gamma$ of the group action on the binary
rooted tree, for $n\leq 12$, are presented. Furthermore, we delve into a
numerical approximation of the spectrum of the discrete Laplacian operator,
which is defined on the limit Schreier graph
$\hat{\Gamma}=\lim_{n\rightarrow\infty} \Gamma_{n}$.
