You need a solvent based dye/stain like a leather stain.
I use Laurel Mtn stock stain. I don't know what their solvent is that they use but it works very well.
Colors can be mixed. Too dark an application can be lightened up with a rub-down with a pad dampened with acetone.
Aqua Forte is another good one that is water based and uses some heat applied to the stained wood to bring out the reddish brown color after application. A favorite of M/Ldr builders.
If mixed wrong and the acid component not completely depleted in doing so, the A/Forte mix can cause some rust problems further down the road with the steel and iron parts inletted into the A/Forte stained wood.
It will also leach the zinc componenet out of the inletted brass furniture on the M/Ldrs leaving the edges of the parts copper colored and the wood bordering the part will show a slight halo of silver. That halo being the zinc.
Brass is a mix of copper and zinc.
Probably not too many brass inlaid stocks on guns built on this forum, but it's worth noting the problem I believe.
Good old oil stain just won't do much on really hard woods like maple, birch, beech and the like. The pores are so small the oil stain 'mud' (the finely ground powdered color held in suspension in the oil carrier) just can't penetrate down into the wood. So you get little color change.
The solvent stains are a solution of the color, like a glass of red wine. It easily soaks in deep.
You won't control the grain pattern of the wood but you can use different colors and shades of them independently to create streaks and faux grain on the wood. This works especially best after the initial staining is done.
Overdone it can look crude. A light touch and the lines feathered out with the above mentioned acetone dampened pad, you can do much to enhance a piece . Sometimes you just want to balance the look.
The solvent stains penetrate so well, you can keep applying them and letting them soak in to dry. Using Ebony, you can make a Birch or maple stock as black as ebony if you want to.