If I understand the question correctly, then $\nu^{2}(1)$ isn't in the image of the natural map $A(G) \to R(G)$ when $G = Q_{8},$ the quaternion group of order $8$.
The number of square roots of the identity in $G$ is $2$, the number of square roots of the central involution $z$ in $G$ is $6$, and the number of square roots of each element of order $4$ is $0$.
Hence we have $\nu^{2}(1) = \lambda_{1} + \lambda_{2}+ \lambda_{3} + \lambda_{4} - \chi$, where the $\lambda_{i}$ are the linear characters of $G$ and $\chi$ is the unique irreducible character of $G$ of degree $2$.
Now I claim that this virtual character is not a difference of permutation characters. Indeed, it is not even a difference of characters afforded by a a virtual $\mathbb{R}G$-module.
For the irreducible character $\chi$ has real Schur index $2$, so occurs with even multiplicity in any character of $G$ afforded by an $\mathbb{R}G$-module.
Hence $\chi$ occurs with even multiplicity in any difference of permutation characters, so that $\nu^{2}(1)$ is not expressible as a difference of permutation characters.
More generally, if $G$ is any finite group of even order which has an irreducible character $\chi$ with Frobenius-Schur indicator $-1$, then $\nu^{2}(1)$ is not expressible as a difference of permutation characters of $G$.
An induction theorem of G. Segal may be relevant to trying to characterize exceptions for which $\nu^{2}(1)$ is not a virtual permutation character of $G$.