Venting basement kitchen through a hot water heater exhaust

Background: we have a "kitchenette" in our basement that only gets occasional use. My lady friend has all sorts of talents, occasionally scorching a casserole is one of them, filling the basement with smoke and setting off the smoke detector. We have an electric oven. There's a pre-existing exhaust for the hot water heater, which is conveniently located near the kitchen area

Question: can I T-Off of the water heater's exhaust, extend it to the wall of the kitchen and tie in a small exhaust fan into the piping leading out the chimney?

My concerns are: a) forced air exhaust negating the chimney effect, b) potential grease laden vapor being exhausted through the chimney c) my basement stinking like burnt bacon