Let $E$ be an elliptic curve. I want to consider its degeneration to the union of two projective lines $C:=\mathbb{P}^1 \cup\_{x,y} \mathbb{P}^1$ attaching at two points $x,y$. The involution $-1$ on $E$ can "degenerates" to an involution on $C$ that preserve each component $\mathbb{P}^1$ and swaps $x$ and $y$. 


My question is, can the translation by a 2-torsion point on $E$ degenerate to an involution on $C$? If this is possible, what kind of action will it be?


More generally is it true that for any translations $t\_1, t\_2$ on $E$, there exists a degeneration $C$ of $E$, such that all $t\_1, t\_2, -1$ degenerate to some action on $C$?



A question on degeneration of elliptic curves with actions. 
Think of the group $E$ as degenerating to the group $C\_{reg} \cong {\mathbb G}\_m \times {\mathbb Z}/2$. In that group, the $2$-torsion is ${\mathbb Z}/2 \times {\mathbb Z}/2$, where one generator rotates each component of $C$ by $-1$, and the other one switches the two components.


EDIT: I'm very disturbed by the fact that $E$ can first degenerate to a one-nodal elliptic
curve, whose group is only ${\mathbb G}\_m$, i.e. with $2$-torsion only ${\mathbb Z}/2$. I retract my claim that one should think of $C\_{reg}$ as a group, instead of a $({\mathbb G}\_m \times {\mathbb Z}/n)$-space, as in S. Carnahan's answer. In that version, it becomes clear that the ${\mathbb Z}/n$ does indeed permute the nodes.


