A projective hypersurface $\mathcal{V}(F)$, given by a homogeneous polynomial $F$, can always be expressed as the union of its affine components $\mathcal{V}(F_{i})$, where $F_{i}=F(x_{1},\ldots,x_{i-1},1,x_{i+1},\ldots,x_{n})$ is the $i$-th dehomogenisation. That is we have $$\mathcal{V}(F)=\bigcup_{i=1}^{n}\mathcal{V}(F_{i}).$$ My question is; is there any relationship between the automorphism group of the projective hypersurface, $\mathrm{Aut}(\mathcal{V}(F))$, and the affine automorphism groups, $\mathrm{Aut}(\mathcal{V}(F_{i}))$?
To me it seems that each automorphism of $\mathcal{V}(F)$ should act either as an isomorphism between pairs of its affine components $\mathcal{V}(F_{i})$ and $\mathcal{V}(F_{j})$, $i\neq j$, or as an automorphism on each $\mathcal{V}(F_{i})$.
So then it should be the case that each $\mathrm{Aut}(\mathcal{V}(F_{i}))$ is a subgroup of $\mathrm{Aut}(\mathcal{V}(F))$?
For generic $F$, $\mbox{Aut}(\mathcal{V}(F))< \mbox{Aut}(\mathbb{P}^{n-1})$ and there is a subgroup of $\mbox{Aut}(\mathcal{V}(F))$ that acts on $\mathcal{V}(F_i)$, namely the stabilizer of $\{x_i=0\}$.
If you can find coordinates such that every atutomorphism in $\mbox{Aut}(\mathcal{V}(F))$ permutes the $\mathcal{V}(F_i)$, then $\mbox{Aut}(\mathcal{V}(F))$ is called an imprimitive subgroup of $\mbox{Aut}(\mathbb{P}^{n-1}$). It can defenitely happen but not always. (For curves it happens for the Fermat curves but not for the Klein quartic)
The problem with $\mbox{Aut}(\mathcal{V}(F_i)) < \mbox{Aut}(\mathcal{V}(F))$ is what you want to the automorphism to preserve. You could have a birational self-map $T$ of $\mathcal{V}(F)$ preserving $\mathcal{V}(F_i)$ and this imply that $T \in \mbox{Aut}(\mathcal{V}(F_i))$ but $T\not\in \mbox{Aut}(\mathcal{V}(F))$.
Let me know if it helps you.