Extending group actions over a codimension two subset Suppose $X$ is a smooth projective variety and $U$ is an open subset whose complement is of codimension two. If a finite group $G$ acts on $U$ does it always extend to $X$ ?
 A: No, it does not always extend.  For instance, let $X\subset \mathbb{A}^4\times \mathbb{P}^1$ be the closed subset of points
$((x_0,x_1,x_2,x_3),[Y_0,Y_1])\in \mathbb{A}^4\times \mathbb{P}^1$ such that $$x_0x_3-x_1x_2 = x_1Y_0-x_0Y_1=x_3Y_0-x_2Y_1=0.$$
This is smooth of dimension 3 (it is one of the two small resolutions of a threefold $A_1$ singularity).  Consider the open subset $U$ where $(x_0,x_1,x_2,x_3)$ is not $(0,0,0,0)$.  The projection $$\text{pr}_{\mathbb{A}^4}:U\to \mathbb{A}^4\setminus\{(0,0,0,0)\},$$
defines an isomorphism of $U$ with the (relatively) closed subset $V$ defined by $x_0x_3-x_1x_2=0$.  In particular, the induced map $$\text{pr}_{\mathbb{P}^1}\circ \text{pr}_{\mathbb{A}^4}^{-1}:V\to U \to \mathbb{P}^1,$$
is the unique morphism such that both $x_1Y_0-x_0Y_1=0$ and $x_3Y_0-x_2Y_1=0$ hold. 
The complement of $U$ has codimension $2$.  Now let $G=\mathbb{Z}/2\mathbb{Z}$ act on $V$ by $(x_0,x_1,x_2,x_3) \mapsto (x_0,x_2,x_1,x_3)$.  Via the isomorphism $U\to V$, this induces an action of $G$ on $U$.  There is no way to extend this to an action on all of $X$.  Essentially this is because there is no action of $G$ on $\mathbb{P}^1$ such that $\text{pr}_{\mathbb{P}^1}\circ \text{pr}_{\mathbb{A}^4}^{-1}$ is $G$-equivariant.
