Question on neutral and stochastic voting rules of variable size

I have a question regarding neutral and stochastic voting rules with domain of variable size.

THE SETUP

Let $$N=\{1,2\}$$ be a two player set, let $$A_\tau=\{a_1,\dots,a_\tau\}$$ be a finite alternative set, where $$\tau\in\mathbb{N}=\{1,2,\dots\}$$; let $$\begin{gather} \mathcal{P}(A_\tau)=\bigl\{P\mid P=(P_i)_{i\in N}:P_i\text{ is voter i's linear order over }A_\tau\bigr\} \end{gather}$$ be the set of all linear order profiles on $$A_\tau$$, let $$\begin{gather} \Delta(A_\tau)=\left\{\delta\in[0,1]^{A_\tau}\mid\sum_{x\in A_\tau}\delta(x)=1\right\} \end{gather}$$ be the set of all lotteries over some alternative set $$A_\tau$$; and let $$\begin{gather} \sigma:\bigcup_{\tau\in\mathbb{N}}\mathcal{P}(A_\tau)\to\bigcup_{\tau\in\mathbb{N}}\Delta(A_\tau) \end{gather}$$ be a voting rule that satisfies $$\sigma(P)\in\Delta(A_\tau)$$ for all numbers $$\tau\in\mathbb{N}$$ and all linear order profiles $$P\in\mathcal{P}(A_\tau)$$.

NEUTRALITY

Given any number $$\tau\in\mathbb{N}$$, let $$M_\tau=\{\mu_\tau\in (A_\tau)^{A_\tau}\mid\mu_\tau\text{ is bijective}\}$$ be the set of all permutations over the alternative set. Then, given any number $$\tau\in\mathbb{N}$$, any strict preference profile $$P\in\mathcal{P}(A_\tau)$$ and any permutation $$\mu_\tau\in M_\tau$$, let the strict preference profile $$\mu_\tau P\in\mathcal{P}(A_\tau)$$ satisfy, for all voters $$i\in N$$ and all alternatives $$x,y\in A_\tau$$, $$\mu_\tau(x)\mu_\tau P_i\mu_\tau(y)$$ if and only if $$x P_iy$$. Further, given any lottery $$\delta\in\Delta(A_\tau)$$ and any permutation $$\mu_\tau\in M_\tau$$, let $$\mu_\tau\delta=(\delta(\mu_\tau(x)))_{x\in A_\tau}$$.

A voting rule $$\sigma$$ is neutral if and only if it is independent of alternatives' labels. Formally, if and only if $$\begin{gather}\label{eq.n} (\forall\tau\in\mathbb{N})(\forall P\in\mathcal{P}(A_\tau))(\forall\mu_\tau\in M_\tau)[\mu_\tau\sigma(P)=\sigma(\mu_\tau P)] \end{gather}$$

THE QUESTION

Consider any neutral voting rule $$\sigma$$. Now, let $$A=\{x,y\}$$ and $$A'=\{w,z\}$$ be two sets of two alternatives each. Further, let $$P=(P_i)_{i\in N}$$ be a linear order profile on $$A$$ such that $$xP_1y$$ and $$yP_2x$$, and let $$P'=(P'_i)_{i\in N}$$ be a linear order profile on $$A'$$ such that $$wP_1'z$$ and $$zP_1'w$$. Namely, $$P'$$ is constructed from $$P$$ by re-naming $$x$$ to $$w$$ and $$y$$ to $$z$$.

Suppose that $$(\sigma(P))(x)=(\sigma(P))(y)=1/2$$. Then, by the neutrality axiom, the voting rule $$\sigma$$ is independent of alternatives’ labels. Thus, $$(\sigma(P'))(w)=(\sigma(P))(x)=1/2$$ and $$(\sigma(P'))(z)=(\sigma(P))(y)=1/2$$.

Let $$A''=\{x_1,x_2,y\}$$ be a set of three alternatives, and let $$P''=(P''_i)_{i\in N}$$ be a linear order profile on $$A''$$ such that $$x_1P''_ix_2P''_1y$$ and $$yP''_2x_1P''_2x_2$$. Namely, for both players $$i\in N$$, $$P’’$$ has been constructed from $$P$$ by replacing $$x$$ for $$x_1P''_ix_2$$. Suppose that $$(\sigma(P''))(\{x_1,x_2\})=1/2$$ and $$(\sigma(P''))(y)=1/2$$.

Then, is it also true that by the neutrality axiom, $$(\sigma(P'))(w)=(\sigma(P''))(\{x_1,x_2\})=1/2$$ and $$(\sigma(P'))(z)=(\sigma(P''))(y)=1/2$$?