Non-normal covering of a Klein bottle by torus. I am trying to construct a non-normal covering of Klein bottle with itself and by a torus.
For Klein bottle to Klein bottle, I got a three sheeted covering, just glue three Klein bottles, which is non-normal but whatever covering I am constructing by torus it is coming normal. Any help will be appreciated. 

Consider the fundamental group of the Klein bottle, $K=\langle x,y\mid y^{-1}xy=x^{-1}\rangle$.
You are looking for a finite index subgroup of $K$ (call it $H$), that is abelian and non-normal. Note that any such covering factors through the orientable double cover, corresponding to the subgroup $\langle x, y^2\rangle$.
Consider $H=\langle xy^{-2}, y^6\rangle$.  This is abelian, non-normal, and of index $6$ in $K$.  It can be described by the diagram below:

To see that this covering is non-normal, here is a picture of the resulting torus (right before the final gluing): the different colored regions are left invariant by the deck transformations (you can also see there are two distinct paths traced by the black arrows, which border the different colors):

