This paper introduces and studies a family of generalized shift-invariant maps \(\chi_{I,J_m}\), which extends the classical KECCAK \(\chi\) mapping. We define an auxiliary family \(\lambda_{h}\) and analyze the collisions of their exponents. A complete criterion is given to decide whether the sequence \(\{\lambda_h\}_{h\ge 0}\) eventually vanishes or becomes periodic; these two regimes are treated separately. In the vanishing case the \(\lambda_h\) become zero after a finite threshold, and the non‑zero ones are linearly independent, spanning a finite‑dimensional vector space. The set \[ G_n=\Bigl\{\mathrm{id}+\sum_{h=1}^{\ell}a_h\lambda_h : a_h\in\mathbb{F}_2\Bigr\} \] forms a finite Abelian group, which is shown to be isomorphic to the unit group of the polynomial ring \(\mathbb{F}_2[x]/(x^{\ell+1})\). This isomorphism yields explicit formulas for the inverse and the order of each element, as well as a description of the iterates of \(\chi_{I,J_m}\) and their fixed points. In the periodic case the family becomes periodic after the index \(q=n/\gcd(n,m)\), and the whole structure reduces to a finite‑dimensional setting. The corresponding group is isomorphic to the multiplicative monoid of \(\mathbb{F}_2[x]/(x^{2q}+x^{q})\), and a concrete criterion for an element to be a permutation is provided. The inverses of these elements can be computed using the extended Euclidean algorithm in \(\mathbb{F}_2[x]\). As an illustration we construct explicit permutations, including the family \(\kappa_{m,I}\).