On the mod-2 cohomology of the product of the infinite lens space and the space of invariants in a generic degree

Let $\mathbb S^{\infty}/\mathbb Z_2$ be the infinite lens space. Denote the Steenrod algebra over the prime field $\mathbb F_2$ by $\mathscr A.$ It is well-known that the cohomology $H^{*}((\mathbb S^{\infty}/\mathbb Z_2)^{\oplus s}; \mathbb F_2)$ is the polynomial algebra $\mathcal {P}_s:= \mathbb F_2[x_1, \ldots, x_s],\, \deg(x_i) = 1,\, i = 1,\, 2,\ldots, s.$ The Kameko squaring operation $(\widetilde {Sq^0_*})_{(s; N)}: (\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_{2N+s} \longrightarrow (\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s)_{N}$ is indeed a valuable homomorphism for studying the dimension of the indecomposables $\mathbb F_2\otimes_{\mathscr A} \mathcal {P}_s,$ It has been demonstrated that this $(\widetilde {Sq^0_*})_{(s; N)}$ is onto. Motivated by our previous work [J. Korean Math. Soc. \textbf{58} (2021), 643-702], this paper studies the kernel of the Kameko $(\widetilde {Sq^0_*})_{(s; N_d)}$ for the case where $s = 5$ and the generic degree $N_d = 5(2^{d} - 1) + 11.2^{d+1}.$ We then rectify almost all of the main results that were incorrect in Nguyen Khac Tin's paper [Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. \textbf{116}:81 (2022)]. We have also constructed several advanced algorithms in SAGEMATH to validate our results. These new algorithms make an important contribution to tackling the intricate task of explicitly determining both the dimension and the basis for the indecomposables $\mathbb F_2 \otimes_{\mathscr A} \mathcal {P}_s$ at positive degrees, a problem concerning algorithmic approaches that had not previously been addressed by any author. Furthermore, this paper encompasses an investigation of the fifth cohomological transfer's behavior in the aforementioned degrees $N_d.$