Regarding a claim about conjugacy of prime ideals in decomposition fields

This question follows from Lemma 6.1.1 from CH 6.1 of Field Arithmetic by Fried and Jarden. It is the subsection on Decomposition groups.'

The following paragraph sets up the notation used.

In the construction of Decomposition groups the chapter starts by defining $R$ to be an integrally closed domain with a quotient field $K$. $L/K$ is a finite Galois extension and $S$ is the integral closure of $R$ in $L$. Let $\mathfrak{p}$ be a prime ideal in $R$ then there exits a prime ideal $\beta$ of $S$ lying over $\mathfrak{p}$. Then the book defines decomposition groups ($D_\beta$) and fields of $\beta$. Finally we have the notation $\overline{L}$ is the quotient field of $S/\beta$ and $\overline{K}$ is the quotient field of $R/\mathfrak{p}$.

Let the decomposition field of $\beta$ be denoted by $L_0$ and let $S_0=S\cap L_0$ and $\beta_0=\beta\cap L_0$.

Now in Lemma 6.1.1 (which is showing that $\overline{L}/\overline{K}$ is normal) we get the following claim.

$\forall\sigma\in G-D_\beta$, if $\sigma^{-1}\beta\cap L_0=\beta_0$ then there exists $\tau\in Gal(L/L_0)$ such that $\tau \sigma^{-1}\beta=\beta$.

They then proceed to show a contradiction and my question is why should such a $\tau$ exist?

I have seen that if there are two prime ideals in $S$ lying over the same prime ideal of $R$ then they are conjugate over $K$. So since we have that both $\beta$ and $\sigma^{-1}\beta$ are prime ideals over $\mathfrak{p}$ they must be conjugate over $K$ so I see why there must exist a $\tau'$ in Gal(L/K) that satisfies the above claim but I do not know how to prove this specific claim. Also I have not been able to use the intersection condition.