Help with venting garage??

[Garthok]

New here - I thought I'd enlist the help of the internet which has been great for my vehicles.
I have a small two car garage that gets way too hot here in Atlanta. I'm trying to find an easy solution to install a quiet exhaust fan/vent. There is already exhaust piping from the kitchen stove out through the garage and vents to the side of the house. The vent is a passive flap that opens when the stove fans are on. Can I tap into this existing tubing with some kind of Y section that has a one way flap on the inside and replace the outside vent with an exhaust fan? And what's to keep the air from being sucked in from the kitchen stove overhead?
My requirements are that the exhaust fan move a good amount of air and is fairly quiet. I plan on putting some adjustable vents in the garage door to allow air flow in when the fan is on. Pictures of the existing setup attached...

Thanks-

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[Chris]

Personally I don't see why not as long as you have a way to stop the stove exhaust from going into the garage. Not sure if there is a code violation or not. I believe they make fans to fit inside those pipes that are quite for when you have a long run to help keep things moving.