Free abelian subgroup of index 2.

Question

Let $G$ be a group with the following presentation $G=gp(x,y \mid x^2=y^2=1)$. I need to know, what further information about $G$ can be derived from knowing that $G$ has a free abelian subgroup of index 2.

Answer 1

No further information, as this group, as noted in the comments, is (isomorphic to) the infinite dihedral group, and thus has a free abelian subgroup $F$ of rank one (isomorphic to the integers, that is) and index $2$, and thus nornal.

This is $\langle x y \rangle$, as $(x y)^x = (x y)^y = y x = (x y)^{-1}$.