# $\mathrm{LMod}(X)$ has finite index in $\mathrm{Mod}(X)$

I am reading about the Birman-Hilden conjecture, specifically the thesis of Rebbeca Winarski and there is some basic point I am not getting.

We consider a covering space $$p: Y \rightarrow X$$ between two connected oriented surfaces. Let $$\mathrm{Mod}(X)$$ be the mapping class group of $$X$$ and $$\mathrm{LMod}(X)$$ be the subgroup of $$\mathrm{Mod}(X)$$ of the isotopy classes of diffeomorphisms that lift to diffeomorphisms of $$Y$$.

$$\mathrm{LMod}(X)$$ must be a finite index subgroup but I do not see why.

So, this is what I found.

As $$p$$ is a covering map between surfaces, it is finite. Suppose that $$p$$ is a $$n$$-fold covering map.

Choose a point $$x\in X$$ and choose $$y\in p^{-1}(x)$$. Then $$p_*(\pi_1(Y,y))$$ has index $$n$$ in $$\pi_1(X,x)$$.

Define $$M$$ to be the set of subgroups having index $$n$$ in $$\pi_1(X,x)$$. This set is finite because $$\pi_1(X,x)$$ is finitely generated.

Let $$\mathrm{Mod(X,x)}$$ be the group of isotopy classes of diffeomorphisms fixing $$x$$ and $$\mathrm{LMod(X,x)}$$ be the subgroup of $$\mathrm{Mod(X,x)}$$ that whose elements lift to elements of $$\mathrm{Mod(Y,y)}$$.

$$\mathrm{Mod(X,x)}$$ acts on $$M$$ by permuting its elements, so there is an homomorphism $$\varphi$$ from $$\mathrm{Mod(X,x)}$$ to $$\Sigma_M$$ (the permutation group on the elements of $$M$$). As $$\Sigma_M$$ is finite, the kernel of $$\varphi$$ has finite index in $$\mathrm{Mod(X,x)}$$.

Now, $$\mathrm{LMod(X,x)}$$ is the stabiliser of $$p_*(\pi_1(Y,y))$$ in $$\pi_1(X,x)$$, so it contains the kernel of $$\varphi$$. This proves that $$\mathrm{LMod(X,x)}$$ has finite index in $$\mathrm{Mod(X,x)}$$.

The rest follows by applying the forgetful map $$\mathrm{Mod(X,x)} \rightarrow\mathrm{Mod(X)}$$, which defines an isomorphism.