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Let $X$ be a smooth projective variety. Then we have an exact sequence:

$$0\mapsto Aut^{o}(X)\rightarrow Aut(X)\rightarrow H\mapsto 0$$

where $Aut^{o}(X)$ and $H$ are respectively the connected component of the identity and the group of the connected components of $Aut(X)$.

Assume that there is a GIT quotient $Y:=X//Aut^{o}(X)$ which is a smooth projective variety as well. Does there exist a morphism of groups $H\rightarrow Aut(Y)$?

Thanks in advance.

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You have not specified anything about the linearizing invertible sheaf $\mathcal{L}$ that you are using to define the GIT quotient. If for every $g\in \text{Aut}(X)$, $g^*\mathcal{L}$ is isomorphic to $\mathcal{L}$, then there is an induced action of $H$ on $Y$. This follows from Section 5, Chapter 1 of "Geometric Invariant Theory" (particularly the last paragraph of the section). It is also fairly easy to see directly from the construction of $Y$ as $\text{Proj}\oplus_n H^0(X,\mathcal{L}^{\otimes n})^G$.

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