$\sigma^{-1} G \sigma$ relationship with $G$ and Aut$(X) \sigma \cap G$

Question

Let Sym$(V)$ is a symmetric group over set $V = \{1,2,3\cdots ,n\}$ and $G$ is a subgroup of Sym$(V)$ i.e. $G \le$ Sym$(V)$. Let $\sigma \in $ Sym$(V)$, then what is $\sigma^{-1} G \sigma$ relationship with $G$.

My questions :

1. Is $\sigma^{-1} G \sigma$ group or coset ?
2. what is the relationship between $G$ and $\sigma^{-1} G \sigma$ ?
3. what is Aut$(X) \sigma \cap G$ (Aut means automorphism group) ?

To me it looks that $\sigma^{-1} G \sigma$ is a group not a coset (because $G\sigma$ is a coset) . But I am not able to prove this formally, please help