<p>Let us consider the prime field of two elements, $\mathbb F_2\equiv \mathbb Z_2.$ It is well-known that the classical "hit problem" for a module over the mod 2 Steenrod algebra A is an interesting and important open problem of Algebraic topology, which asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$ on $m$ variables $x_1, \ldots, x_m,$ each of degree one, regarded as a connected unstable $\mathscr A$-module The algebra $\mathcal

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... _m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan complex $K(\mathbb F_2, 1).$ Although the hit problem has been thoroughly studied for more than 3 decades, solving it remains a mystery for any $m\geq 5.$ In this article, we develop our previous work [Commun. Korean Math. Soc. 35 (2020), 371-399] on the hit problem for $\mathscr A$-module $\mathcal P_5$ in generic degree $n:=n_s = 5(2^{s}-1) + 18.2^{s}$ with $s\geq 0.$ As an immediate consequence, a local version of Kameko's conjecture for the dimension of the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ associated with weight vectors is true for $m = 5$ and degree $n_s.$ Also, we show that this conjecture also holds for any $m\geq 1$ and degrees $\leq 12.$ Two applications of this study are to establish the dimension result for the cohit space $\mathbb F_2\otimes_{\mathscr A}\mathcal P_m$ for $m = 6$ in generic degree $5(2^{s+4}-1) + n_1.2^{s+4}$ with $s > 0$ and to describe the modular representations of the general linear group of rank $5$ over $\mathbb F_2.$ Our results then show that the algebraic transfer, defined by W. Singer [Math. Z. 202 (1989), 493-523], is an isomorphism in bidegrees $(5, 5+n_s)$ with $s\geq 0.$ Also, we prove that Singer's transfer is a trivial isomorphism in bidegrees $(m, m+12)$ with $m\geq 1.$</p>