gr.group theory - Brauer's permutation lemma -- extending to some other finite groups?

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Let $G$ be a finite subgroup of $GL(V)$. I claim that the following are equivalent:

$(1)$ Any two elements of $G$ which are $GL(V)$ conjugate are also $G$ conjugate.

$(2)$ The representation ring $mathbb{Q} otimes mathrm{Rep}(G)$ is spanned by representations $S^{lambda}(V)$, where $S^{lambda}$ ranges over all Schur functors.

Proofs: For $g$ in $G$, let the eigenvalues of $g$ be $e^{2 pi i q_k(g)}$, where $(q_1(g), q_2(g), ldots, q_n(g))$ is a multiset of elements of $mathbb{Q}/mathbb{Z}$. Write $Q(g)$ for this multiset. So $g$ and $g'$ are $GL(V)$ conjugate if and only if $Q(g) = Q(g')$. When necessary, we will write $Q_W(g)$ for the analogous construction for some other representation $W$.

If $Q(g)=Q(g')$ then $Q(g^k) = k Q(g) = k Q(g') = Q((g')^k)$. So, for every Adams operator $psi^k$, the actions of $g$ and $g'$ on the virtual representation $psi^k(V)$ have the same eigenvalues and, in particular, the same trace. Also, $Q_{V_1 oplus V_2}(g)$ and $Q_{V_1 otimes V_2}(g)$ are determined by $Q_{V_1}(g)$ and $Q_{V_2}(g)$. So, if $Q(g)=Q(g')$ then the linera functions $Tr(g)$ and $Tr(g')$ are equal on the sub-$Lambda$-ring of $mathbb{Q} otimes mathrm{Rep}(G)$ generated by $V$.

Assume condition 2, and let $Q(g) = Q(g')$. Then $Tr(g)$ and $Tr(g')$ are equal on all of $mathbb{Q} otimes mathrm{Rep}(G)$, so $g$ and $g'$ are $G$ conjugate.

Conversely, assume condition 1. Fix any $h in G$. Let $S$ be the sub-$Lambda$-ring of $mathbb{Q} otimes mathrm{Rep}(G)$ generated by $V$. We will construct a virtual representation $W$ in $mathbb{C} otimes S$ such that $Tr_W(g)$ is $0$ for $g$ not conjugate to $h$ and $1$ for $g$ conjugate to $h$. Taking such linear combinations of such linear functionals, we can clearly generate all class functions on $G$, so $S$ must be the whole of $mathbb{Q} otimes mathrm{Rep}(G)$.

By condition (1), $Q(h)$ is different from $Q(g)$ for any $g$ not conjugate to $h$. Therefore, we can find a symmetric polynomial $F$, with coefficients in $mathbb{C}$, which vanishes at $e^{2 pi i Q(g)}$ for $g$ not conjugate to $h$, but does not vanish at $e^{2 pi i Q(h)}$. (Here $e^{2 pi i Q(g)}$ is a point of $mathbb{C}^n$, which should be considered as defined only up to the action of $S_n$.) Using the standard relation between symmetric polynomials and $Lambda$-ring operations, $F(V)$ is the desired $W$.