Beech stocks on cheap guns were very common on this side of the pond in the past, I will also say that to colour them Walnut is simple but what is incredibly difficult is to make them look like Walnut. Both Beech and Birch timbers have convoluted grain that absorbs colour at differing rates leading them to look blotchy. To minimize this effect I always used spirit or Alcohol base and wood dye crystals, usually a mixture of Walnut and Vandyke Brown crystals. Finish the wood in your usual manner then after the final fine sanding stage burnish the wood hard with a leather pad in an endeavor to close up the more open sections of grain to slow down the rate of colour absorption.
To further slow down colour absorption saturate the woods surface with Spirit or Alcohol then while still wet apply your stain, usually mixing up a number of differing strengths and starting with the weakest first. Continue with wetting the surface with with Spirit or Alcohol then adding more stain stopping when you achieve the best colour compromise, remember a little colour goes a long way. Try to keep in mind the old saying "you cant make a silk purse out of a Sows ear" but you can have a dam good try.
