For a well pointed co-\(H\)-space \(X\), let \(J(X)\) denote the James construction (i.e. reduced product) of \(X\), and let \(J_n(X)\) be the \(n\)-th stage of the James filtration of \(J(X)\). Since there is the James filtration \(X=J_1(X)\subset J_2(X)\subset \cdots \subset J_n(X)\subset J_{n+1}(X)\subset \cdots\) and \(J(X)=\varinjlim J_n(X)\), the total Cohen group \([J(X),\Omega Y]\) is represented as \([J(X),\Omega Y]=\varprojlim [J_n(X),\Omega Y].\) In this paper, the authors study the co-\(H\)-structure of the generalized Fox space \(\hat{\mathfrak{F}}(X)\) via the generalized Whitehead products and they investigate the reduced suspension co-\(H\)-structure on \(\Sigma J_n^*(X)\), where \(J_n^*(X)=J(X)\sqcup *\). As an application they determine the group structure of \([J_n(X),\Omega Y]\) and \([J(X),\Omega Y]\) by using the argument given in their paper [Q. J. Math. 66, No. 1, 111–132 (2015; Zbl 1312.55012)] replacing \(S^1\) by a co-\(H\)-space \(X\). Moreover, they obtain an example that \([J(S^3),\Omega S^{10}]\) is not a subgroup of \([J(S^1),\Omega S^{10}]\), and they also construct a space \(\hat{Y}\) such that \(\Omega \hat{Y}\) is not homotopy commutative and the group \([J(S^r),\Omega \hat{Y}]\) is abelian for any \(r\geq 1\).