Stan, do you think it is a matter of fit? 
 I shot a splinter/straight grip, plastic buttplate, 12 yesterday and noticed no movement of the gun in my hands at firing.  I don't shoot more than 1 1/8.
They were 1200+fps though.
I didn't notice any muzzle flip either. Managing to stay on line with some true pairs pretty well. 
I control the line of the barrels with my left hand, my right just brings the stock up to my face, and back into my shoulder pocket.  I don't like feeling the buttstock squirming back into my shoulder pocket, it ends up in barrel checking.  Messes up my timing.