Kronecker product preserves the conjugacy relation?

Question

Let $G =$ PGL$_{n}(\textbf{C})$ and $T$ be the image in $G$ of the subgroup of the invertible diagonal matrices of $\operatorname{GL}_{n}(\textbf{C})$. Let $A$ and $B$ be two elementary abelian $2$-subgroups in $T$ of the same rank.

The Kronecker product $\otimes I_{2}$ embeds $A$ and $B$ in $H$ = $\operatorname{PGL}_{2n}(\textbf{C})$. If $A$ and $B$ are not conjugate in $G$, will $A\otimes I_{2}$ and $B\otimes I_{2}$ not be conjugate in $H$? Intuitively, they are not conjugate in $H$ but I'm not sure of tools to tackle it. Maybe subgroup conjugacy is an equivalent relation and $\otimes I_{2}$ reserves it?

Edit: Some thinking made based on the input in the comment. If $A$ and $B$ are not conjugate in $G$, and we assume $A\otimes I_{2}$ and $B\otimes I_{2}$ are conjugate, attempt to get a contradiction. Since $A\otimes I_{2}$ and $B\otimes I_{2}$ are both block diagonal matrices with blocks $I_{2}$ and $-I_{2}$, if there always exists a "block permutation matrix" with each block $I_{2}$, then we're done. But I'm not sure about the existence of such a "block permutation matrix"... Any hints would be appreciated.

Answer 1

If $A$ and $B$ are elementary abelian $2$-subgroups of $\mathrm{PGL}_n(\mathbf C)$ of rank $r$ then they lift uniquely to elementary abelian $2$-subgroups of $\mathrm{GL}_n(\mathbf C)$ of rank $r+1$ (take all lifts $a$ of elements of $A$ such that $a^2 = 1$), so let us assume $A, B \le \mathrm {GL}_n(\mathbf C)$ to begin with.

Consider a representation $\rho:C_2^r \to \mathrm{GL}_n(\mathbf C)$ with character $\chi$. Decompose $\chi$ into irreducible characters: $\chi = w_1 \chi_1 + \cdots + w_m \chi_m$. Here $\chi_1, \dots, \chi_m : C_2^r \to \{\pm1\}$ and $w_1, \dots, w_m$ are positive integers. Two representations $\rho$ and $\rho'$ are equivalent (conjugate) if and only if they have the same character, so equivalence classes of representations $\rho:C_2^r \to \mathrm{GL}_n(\mathbf C)$ are in bijection with maps $w: D \to \mathbf N = \{0,1,2,\dots\}$ such that $\sum_{\chi \in D} w(\chi) = n$, where $D \cong C_2^r$ is the dual group of $C_2^r$ (the group of characters $\chi : C_2^r \to \{\pm1\}$ with pointwise product as the group operation). Here $w(\chi_i)$ records the multiplicity of an irreducible character $\chi_i$ in $\chi$. The representation is faithful if and only if the support $\mathrm{supp}(w) = \{\chi : w(\chi) > 0\}$ of $w$ generates $D$.

To get conjugacy classes of embedded copies of $C_2^r$ we must further quotient $\{w:D \to \mathbf N \mid \langle \mathrm{supp}~w\rangle = D\}$ by the action of $\mathrm{Aut}(C_2^r) \cong \mathrm{GL}_r(2)$.

In any case the Kronecker product just sends $w$ to $2w$, so the conjugacy relation is preserved.