  In this paper we present the perturbative evaluation of the difference
between the renormalization functions of flavor singlet and nonsinglet bilinear
quark operators on the lattice. The computation is performed to two loops and
to lowest order in the lattice spacing, for a class of improved lattice
actions, including Wilson, tree-level (TL) Symanzik and Iwasaki gluons, twisted
mass and SLiNC Wilson fermions, as well as staggered fermions with twice
stout-smeared links. In the staggered formalism, the stout smearing procedure
is also applied to the definition of bilinear operators.
