Conjugate upto isomorphism Let $V= \{1,2,3,\cdots, n\}$ is a vertex set and Sym$(V)$ is a symmetric group. Let $G \le $ Sym$(V)$, i.e. subgroup of Sym$(V)$. Now 
$$H= Sym(V_0^{\sigma}) \times Sym(V_1^{\sigma}) \cdots \times Sym(V_n^{\sigma}) $$
where $\sigma \in$ Sym$(V)$ and $V_i$ means set of vertices of degree $i.$
My question : What is a relationship between $G$ and $\sigma^{-1}H\sigma$ ? To me it appears that they are conjugate to each other upto isomorphism. 
Second question : Is it true that $x^g = x^{\sigma^{-1}g_1 \sigma}$, where $g \in G$ and $g_1 \in H$ ?
 A: In chat you explain that $V_i^{\sigma}$ is supposed to be the image of $V_i$ under the function $\sigma_i$. This is contrary to standard notation in the context of group actions: the standard notation is $gX$ for the image of $X$ under $g$, and $X^g$ for the set of fixed points of $g$ in $X$.
The question is nonsense without saying what $G$ is. Indeed, the obvious interpretation is to say $G$ is the automorphism group of the graph. ($V$ is the vertex set of some graph.)
In this case, since graph automorphisms preserve vertex degree, we must have
$$ G\subseteq \mathrm{Sym}(V_0)\times\mathrm{Sym}(V_1)\times\cdots\times\mathrm{Sym}(V_n)\subseteq\mathrm{Sym}(V) $$
where the product is internal (so $\mathrm{Sym}(V_i)$ is the subgroup of $\mathrm{Sym}(V)$ of permutations whose support is contained in the complement $V\setminus V_i$).
Then
$$ \begin{array}{lcccccccc} H & = &  \mathrm{Sym}(\sigma V_0)~ & \times &  \mathrm{Sym}(\sigma V_1)~ & \times & \cdots & \times & \mathrm{Sym}(\sigma V_n)~ \\ & = & \sigma \mathrm{Sym}(V_0)\sigma^{-1} & \times & \sigma\mathrm{Sym}(V_1)\sigma^{-1} & \times & \cdots & \times & \sigma\mathrm{Sym}(V_n)\sigma^{-1} \end{array} $$
$$ =\sigma\big(\,\mathrm{Sym}(V_0)\times\mathrm{Sym}(V_1)\times\cdots\times\mathrm{Sym}(V_n)\, \big)\sigma^{-1} $$
contains $\sigma G\sigma^{-1}$. This containment follows from conjugating $G\subseteq\prod_i \mathrm{Sym}(V_i)$ by $\sigma$.
Note the containment is proper unless $G=\prod_i\mathrm{Sym}(V_i)$, which pretty much never happens.
