The ordinary cohomology transfer associated to the orbit space projection of a finite group action need not commute with stable cohomology operations. In particular, if an even group acts on a space, the resulting transfer τ will not generally commute with the Steenrod squares, Sqi. This paper contains a formula for the deviation from commutativity (Sqiτ — τSqi)χ in the case of an involution. The formula involves the restriction of χ to the cohomology of the fixed point set, as well as certain naturally occurring characteristic classes.