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.



Kronecker product preserves the conjugacy relation?
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.


