Basement insulation

My basement was finished but the walls were covered with 1/8 plywood panels, I wanted to insulate the outside walls so I bought Dow rigid foam 1" thick, 4X8 panels. When I took out the plywood, there was 1/4 drywall panels, installed over furring, and the wall is covered with black paper vapor barrier . The furring attached to the wall is 1X6 horizontal top and bottom, and 1X2 verticaly. Real thickness is 3/4. Not all the furring is perfect secured to the wall (some areas are lose), and the vertical stripes are not equally distanced. Also seems like there is some slope on the upper portion of the wall as the vertical furring is not aligned with top horizontal. I understood in order to install the boards I have to take out the vapor barrier and attach the foam direct to the wall (I will use construction foam glue). I ask what should you suggest I do from below 3 options, or other suggestions.
1. Secure current furring, and cut the boards to fill the empty space between the wood. As foam is is 1"and furring is 0.75", I have to add everywhere plywood strips in top of current furring, and only after that to put drywall.
2. Remove all furring. Attach continuously the boards to the wall, and install vertical furring 16" distanced, and top and bottom planks, using the one removed or buy new ones (1X2). I don't want to be far from foam to drywall. 3. Remove all furring, frame the wall with 2X2 studs, and install the boards between the studs.
My intention is to secure the studs with tapcon screws. By the way if I remove the furring, should I fill the remaining holes from removed nails, or doesn't matter as will be covered by foam?

 

Remove the furring, you want as close to a continuous wall of foam as possible. It's generally more common to then build a 2x4 stud wall, which can then contain fiberglass or mineral wool insulation to further build the R value.

 

Thank you for answer. I don't know How I would attach a 2X4 sideway to the wall (long screws or counter bore). However if I would add another 2" in top of 3/4 by changing the 1" furring with 2X2, not only I'm losing a bit of room space (less concern), but I would have to readjust the frame of a window and a patio door (walk out basement).
Anyhow if I understood correct, you suggest to attach the foam and put the studs/furring on top, and not making a new frame and install boards in between.
Not sure what dammage to the wall I get when removing the existing furring, but do you think should I patch the holes first, and what material should I use for patching?

 

if you are concerned with the old holes you can use some concrete filler in a caulking gun or hydraulic cement...

what stickshift recommended is the correct way to do it... by removing the old furring strips and running the foam seamlessly, you don't run the risk of any leak behind the current strips that are there... you have a full thermal break of 1" foam behind the board... if you are using tapcons you can just lay the furring strips over the foam and then secure straight into the concrete... if you want more insulation, you can then put 1/2" or 3/4" foam in between your furring strips so then you have a total 1.5-1.75" foam before sheetrock...

here in the states we have something called INSULPINK... it has channels on each end preformed so you can lay 1x3 furring strips flush to the top of the foam... then sheet rock on top... this is useful when you can't afford to lose a few more inches of space by building a traditional stud wall on top of the foam...

 

Not sure if I was clear about this but I would build a normal stud wall which is attached to the floor and ceiling (not the outside wall) adjacent to the foam on the outside wall.

 

i think he said that he wanted to be as tight to the wall as possible so that he wouldn't have to reframe a door and window...

the traditional 2x4 frame that your recommended, ramset/tapcon to the floor and nailed to the ceiling joist is the best...

my next house with 8' basement ceilings and rigid foam insulation on the outside of the foundation will have this!