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Let $P_s:= \mathbb{F}_2[x_1,x_2,\ldots ,x_s] = \bigoplus_{n\geqslant 0}(P_s)_n$ be the polynomial algebra viewed as a graded left module over the mod 2 Steenrod algebra, $\mathscr A$. The grading is by the degree of the homogeneous terms $(P_s)_n$ of degree $n$ in the variables $x_1, x_2, \ldots, x_s$ of grading $1$. We are interested in the {\it hit problem}, set up by F. P. Peterson, of finding a minimal system of generators for $\mathscr A$-module $P_s$. Equivalently, we want to find a basis for the $\mathbb F_2$-graded vector space $\mathbb F_2\otimes_{\mathscr A} P_s$. In this paper, we study the hit problem in the case $s=5$ and the degree $n = 5(2^t-1) + 6\cdot 2^t$ with $t$ an arbitrary positive integer.
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